Chimera: A Massively Parallel Code for Core-collapse Supernova Simulations

For multidimensional applications, Chimera uses the method of dimensional splitting (Strang 1968). In this method, the numerical solution of the multidimensional problem proceeds by a series of one-dimensional steps or "sweeps" to build up the full solution. The updated variables after each sweep are used as initial conditions for the next sweep. The order of the sweeps is shown in Figure 1. For second-order accuracy in time, the time step, Δt, should be selected at the beginning of the sweep sequences, and for the 2D case, $\tfrac{1}{2}{\rm{\Delta }}t$ should be used in the x-y subsequence of the sweep sequence and again in the y-x subsequence. Likewise, in the 3D case, $\tfrac{1}{4}{\rm{\Delta }}t$ should be used for each of the four x-y-z transposition subsequences. In practice, the full time step computed before each subsequence is used for that subsequence. Since the time steps computed for each of the subsequences are nearly the same, approximate time-centering is maintained. Furthermore, the larger time steps permit a more refined grid to be used for the same amount of computer time.

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Chimera is designed to numerically evolve CCSNe on spherical polar grids and to run on multiprocessor machines. The domain decomposition in Chimera is dictated, in part, by the present implementation of neutrino transport, which is in the radial direction. The decomposition is along rays, or bundles of rays, each ray being one zone wide in two dimensions and spanning the entire set of zones from boundary to boundary along the third. Thus, an x-ray, or radial ray refers to a set of zones along a radial line from the center to the outer edge of the grid. A y-ray, or angular ray, refers to a set of zones at a given radius and azimuth which, for a 180° angular grid, spans the arc from the "north" pole to the "south" pole. Finally, a z-ray, or azimuthal ray, refers to a set of zones at a given radius and polar angle that, in the case of a 360° azimuthal grid, completely encircles the polar axis.

As an extremely simple example of the domain decomposition in Chimera, consider a 2D grid consisting of four radial rays, each radial ray consisting of 12 radial zones with one radial ray per MPI rank, four MPI ranks in all. Figure 2 shows the logical structure of this grid. The large rectangular block bordered in thick black lines encompasses the logical grid with the x (radial), y (angular), and z (azimuthal) directions shown at the upper right. Because the grid is two-dimensional, the third or z-dimension is superfluous but is shown as an elongated cell wall in the z-direction for the sake of comparison with a 3D example below. Each zone is shown in the figure as a square-sided vertical rectangular volume. The total number of radial, angular, and azimuthal zone-centers are denoted by imax, jmax, and kmax, respectively, and the total number of zone edges by ${\mathtt{imax}}+1$, ${\mathtt{jmax}}+1$, and ${\mathtt{kmax}}+1$. In this case, ${\mathtt{imax}}=12$, ${\mathtt{jmax}}=4$, and kmax = 1.

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The computation in Chimera is directionally split, and the (x-, y-, z-) sweeps—i.e., the (radial-, angular-, azimuthal-) sweeps—refer to the direction of computation. During the x-sweep, a set number, or bundle, of radial rays (in this example just one) is assigned to each MPI rank. In Figure 2, a radial ray is shown in green for a particular, but otherwise arbitrary, MPI rank. The dimensions ij_ray_dim and ik_ray_dim denote the dimensions of a bundle of x-rays in the y- and z-directions, respectively, so that ij_ray_dim × ik_ray_dim is the number of x-rays in the bundle. In this example, where the bundle of rays per MPI rank consists of just one ray, both ij_ray_dim and ik_ray_dim are unity. The local indices ij_ray and ik_ray locate a particular ray within a bundle relative to the upper left corner of the bundle, so that ij_ray ∈ [1, ij_ray_dim] and ik_ray ∈ [1, ik_ray_dim]. In this example, both of these indices are unity as there is only one ray per bundle. During the x-sweep, the radial hydrodynamics, radial RbR transport, and nuclear reactions are evolved along with global gravity solves.

Following the x-sweep, a transpose to the y- or z-oriented rays is performed. In this 2D example, the transpose is just to the y-oriented rays. Because there are 12 radial zones and only four angular zones in this example, each MPI rank now consists of a bundle of three y-rays, so that the total number of MPI ranks remain the same. This is delineated in Figure 2 by the rays enclosed by red for a particular, but arbitrary, bundle. Now, (ji_ray, jk_ray) takes the place of (ij_ray, ik_ray) and locates a particular angular ray in the x–z plane of the bundle. The widths in the x–z plane of each bundle are given by j_ray_dim and ik_ray_dim, which in this example are equal to 3 and 1, respectively. j_ray_dim and k_ray_dim are the number of radial zones on each MPI rank after transposing to the y-oriented rays and z-oriented rays, respectively.

Figure 3 illustrates a slight variation of the previous domain decomposition example using the same logical grid. The total number of radial, angular, and azimuthal zones are the same as before, but in this example, bundles of two x-rays are associated with each MPI rank, two MPI ranks in all. In Figure 3, a particular, but arbitrary, bundle of x-rays is shown bounded by thick green lines. In this case, the y–z dimensions of the bundle are given by ij_ray_dim = 2 and ik_ray_dim = 1, respectively, and the particular, but arbitrary, radial ray designated by the narrow-lined green X at the front and top has the local indices in the bundle (ij_ray = 2, jk_ray = 1). Following the x-sweep, a transpose to the y-oriented rays is made, and because there are only two MPI ranks in this case, the number of y-rays associated with each MPI rank is six, as there are 12 y-rays in all. This is delineated in Figure 3 by the rays enclosed by the thick red lines for a particular, but arbitrary, y-bundle, The x–z dimensions of the y-bundles are j_ray_dim = 6 and ik_ray_dim = 1, respectively, and the particular, but arbitrary, y-ray designated by the narrow-lined red X at the front and top has the local indices in the bundle (ji_ray = 5, jk_ray = 1).

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Figure 4 illustrates a general domain decomposition example using the same logical grid for the x- and y-rays, but adding a z-ray consisting of 6 zones. Now there are ${\mathtt{imax}}\times {\mathtt{jmax}}\times {\mathtt{kmax}}=12\times 4\times 6=288$ zones in all, and ${\mathtt{jmax}}\times {\mathtt{kmax}}=4\times 6=24$ x-rays. Let us suppose we wish to compute with 4 MPI ranks. We must then use 4 bundles of x-rays, each bundle assigned to one MPI rank and consisting of 6 x-rays. One such particular, but arbitrary, bundle is shown in Figure 4, outlined in the thick green lines. The y-z dimensions of the bundle are ij_ray_dim = 2 and ik_ray_dim = 3, respectively. A particular, but arbitrary, x-ray is shown in this bundle by a thin-lined green X on its front face, having the local indices (ij_ray = 1, ik_ray = 2); another x-ray located in an adjacent bundle is delineated by another thin-lined green X on its front face having the local indices (ij_ray = 2, ik_ray = 3). On executing the x-sweep, each MPI rank performs the computation required to complete the individual x-sweep for each x-ray in its bundle.

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Following the x-sweep, a transpose to the y- and z-oriented rays is performed, either in that order, or reversed order, as shown above in Figure 1. Consider the transpose to the y-oriented rays. There are ${\mathtt{imax}}\times {\mathtt{kmax}}=12\times 6=72$ y-rays. With four MPI ranks, we assign to each MPI rank four bundles of y-rays with 18 y-rays each, having the x–z dimensions j_ray_dim = 6 and ik_ray_dim = 3, respectively. A particular, but arbitrary, bundle of y-rays is shown bounded by thick red lines in Figure 4, and a particular, but arbitrary, y-ray in that bundle is singled out by a thin-lined red X, having the local indices (ji_ray = 4, jk_ray = 3).

Following or preceding the y-sweep, a transpose to the z-oriented rays is performed. There are ${\mathtt{imax}}\times {\mathtt{jmax}}\,=12\times 4=48$ z-rays. With four MPI ranks, we assign to each MPI rank four bundles of z-rays with 12 z-rays each, having the x–y dimensions k_ray_dim = 4 and ij_ray_dim = 2, respectively. A particular, but arbitrary, bundle of z-rays is shown bounded by thick blue lines in Figure 4, and a particular, but arbitrary, z-ray in that bundle is singled out by a thin-lined blue X, having the local indices (${\mathtt{ki}}\_{\mathtt{ray}}=3,{\mathtt{kj}}\_{\mathtt{ray}}=2$).

In general, for a 2D or 3D grid with a domain decomposition consisting of x-rays, y-rays, and z-rays, to ensure load balancing, the number number of rays in x-bundles, y-bundles, and z-bundles must satisfy

$$\begin{eqnarray}\begin{array}{rcl}{N}_{\mathrm{MPI}} & = & \displaystyle \frac{{\mathtt{jmax}}\times {\mathtt{kmax}}}{{\mathtt{ij}}\_{\mathtt{ray}}\_{\mathtt{\dim }}\times {\mathtt{ik}}\_{\mathtt{ray}}\_{\mathtt{\dim }}}=\displaystyle \frac{{\mathtt{imax}}\times {\mathtt{kmax}}}{{\mathtt{j}}\_{\mathtt{ray}}\_{\mathtt{\dim }}\times {\mathtt{ik}}\_{\mathtt{ray}}\_{\mathtt{\dim }}}=\displaystyle \frac{{\mathtt{imax}}\times {\mathtt{jmax}}}{{\mathtt{ij}}\_{\mathtt{ray}}\_{\mathtt{\dim }}\times {\mathtt{k}}\_{\mathtt{ray}}\_{\mathtt{\dim }}}\end{array}\end{eqnarray} \tag{ 1 }$$

where N_MPI is the number of MPI ranks.

The input and output (IO) of data by simulation codes presents a significant challenge, and good IO is required to achieve acceptable computational performance. Chimera has several components to its IO subsystem. A typical full explosion model requires ${ \mathcal O }(1000)$ hours to complete. This requires writing checkpoint files for restarting the simulation and data files for later analysis, plotting, and visualization. We have implemented two schemes for general restart and analysis IO: (1) the original, serial method used for much of our early 2D work and (2) a parallel method required for effective computation in 3D.

2.3.1. Serial IO Method

2.3.2. Parallel IO Method

In addition to the IO of grid-based data, Chimera also outputs data for passive Lagrangian tracer particles. These are used to record the thermodynamic and neutrino exposure histories of individual mass elements that are then used for post-processing analysis. Following each directional sweep of the hydrodynamics, the position of a tracer particle in that direction at time tⁿ and position $({r}^{n},{\theta }^{n},{\phi }^{n})$ is advanced to ${t}^{n+1}$ according to the simple Euler method:

$$\begin{eqnarray}&&{r}^{n+1}={r}^{n}+{u}_{r}^{n}{\rm{\Delta }}{t}^{n},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{\theta }^{n+1}={\theta }^{n}+\displaystyle \frac{{u}_{\theta }^{n}}{r}\,{\rm{\Delta }}{t}^{n},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\phi }^{n+1}={\phi }^{n}+\displaystyle \frac{{u}_{\phi }^{n}}{{\left(r\sin \theta \right)}^{n}}\,{\rm{\Delta }}{t}^{n},\end{eqnarray} \tag{ 4 }$$

assuming constant velocity $({u}_{r}^{n},{u}_{\theta }^{n},{u}_{\phi }^{n})$ through the time interval ${\rm{\Delta }}{t}^{n}={t}^{n+1}-{t}^{n}$. For the B-series models, which were all 2D and included 4000–8000 particles, individual output files were maintained for each tracer. These contained the physical quantities of interest, linearly interpolated in radius to the tracer particle positions from the zone-center (cell-averaged) values of the computational grid; the lone exception being the interpolation of differential neutrino number fluxes, which are defined at radial zone edges. Output for individual particles was performed whenever the physical quantities of interest changed by 10% (see Harris et al. 2017, for more details).

For the C-series and later models, which included 3D models with as much as 400,000 tracers, this approach proved impractical; thus, Chimera simply records the tracer positions in "Frames" and "Restart" files, with interpolation of quantities of interest relegated to a post-processing step. Nucleosynthesis tests by Harris et al. (2017) show that output intervals of Δt ∼ 1 ms are sufficient in CCSN to resolve the features in the thermodynamic profiles needed for accurate post-processing of the Lagrangian tracers. Thus, the computational time expended on the temporally detailed individual particle traces written out in the B-series models can now be better spent elsewhere.

To test the scalability of Chimera, we computed 3D models with all of the standard physics of the B- and C-series, started from the same 15 ${M}_{\odot }$ progenitor used in the B15-WH07 run (Bruenn et al. 2013, 2016), with 512 radial zones. Models were run from the onset of collapse for a couple of hours for n_θ of 32, 64, 128, and 256, with n_ϕ = 2n_θ, on the Cray XK7 ("Titan") at OLCF, utilizing "maximal decomposition" with one radial ray per process (MPI rank). The results plotted in Figure 5 use the version of the code used for the C15-3D model (Lentz et al. 2015) and show slightly more than doubled wall time for the largest model relative to the smallest, but the typical size of our actual production 3D runs are closer to the 32768-ray (128 × 256) model, which takes about 25% longer than the smallest model to complete 100 steps.

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The lettered series of Chimera simulations are not in the strict sense of software engineering and development practices "versions" of the code, but rather represent a combination of default physics, parameter choices, and a common code base used in those simulations. As it has occurred, all of the series discussed in this paper have utilized different code bases, with progressive improvement with each subsequent series, but in the future, we may use the series designation to separate groups of simulations with different defaults run with the same code. The series label combined with a basic description of the progenitor and any deviations from the series defaults are used to construct unique simulation designations to identify a model across publications.

Initially, there were separate 2D and 3D branches of the code and both were used for the simulations retroactively designated "Series A" (Bruenn et al. 2009) with the 2D branch using only the binary IO for restart and plotting and the 3D branch using HDF5 for restarts. The Series-B runs included many improvements to the code and simulation design, reflecting the acquired experience of Series A. While the 2D-only Series B code was running, the 3D code was optimized and became the only branch of Chimera for all subsequent runs of any dimension and included changes to the code structure needed for the Yin-Yang grid. The Yin-Yang mesh was not yet satisfactory when Series C was started and the major differences between that and Series D are the finalized Yin-Yang grid and an improved radial-mesh motion routine. Major improvements were made to the handling of the NSE state and the boundary with the non-NSE material handled by the network before Series-B, with geometric upgrades before Series D permitting free islands of NSE or non-NSE material to embed, as necessary, in the other. Some operational choices in the dual EoS scheme were made for Series B, while Series E added pre-tabulated equations of state (R. E. Landfield et al. 2020, in preparation). The temporal sub-cycling of the non-radial sweeps deep in the core generated entropy and was replaced for Series B by a frozen core, wherein the non-radial hydrodynamics was skipped, which was in turn replaced by a spherical averaging scheme late in Series B and used in all later series.

In the list that follows, we describe major differences between the code revisions used in the series indicated. For descriptions of the utilized physics and default choices for resolution, the associated simulation papers should be referenced.

• 1.  
  Series A

  • (a)  
    EoS:LS180, ρ > $1.7\times {10}^{8}\,{{\rm{g\; cm}}}^{-3}$
  • (b)  
    Hydrodynamics: subcycle lateral sweep
  • (c)  
    NSE: spherical boundary
  • (d)  
    IO: binary restart and plot files
• 2.  
  Series B

  • (a)  
    EoS: LS220, ρ > $1\times {10}^{11}\,{{\rm{g\; cm}}}^{-3}$
  • (b)  
    Hydrodynamics: frozen core, carbuncle update
  • (c)  
    NSE: 17-species (α, n, p, ${}^{56}\mathrm{Fe}$), boundary a function of latitude
• 3.  
  Series C

  • (a)  
    Opacities: shared points (Section 8.11.5)
  • (b)  
    Hydrodynamics: averaged core
  • (c)  
    Transport: shock fix (Section 6.12), Minerbo closure replaces geometric closure
  • (d)  
    IO: merged restart and plot quantities into single HDF5 file
  • (e)  
    Mesh: Yin-Yang 3D grid-related changes

Two general cases must be distinguished when considering the thermodynamic state of the fluid, NSE, and non-NSE. In NSE, the thermodynamic state of the fluid in a given zone is specified by the values of its density, ρ, temperature, T, and electron fraction, ${Y}_{{\rm{e}}}$. To accommodate the demand for frequent EoS interrogations and the need for derivatives of some of the dependent thermodynamic quantities, Chimera constructs a thermodynamic grid in ($\mathrm{log}\rho ,\mathrm{log}T,{Y}_{{\rm{e}}}$)-space defined by a user specified number, ${\mathtt{dgrid}}({\mathtt{m}})$, of evenly spaced points of $\mathrm{log}\rho$ per decade change in ρ, a user specified number, ${\mathtt{tgrid}}({\mathtt{m}})$, of evenly spaced points of $\mathrm{log}T$ per decade change in T, and a user specified number, ${\mathtt{ygrid}}({\mathtt{m}})$, of points in ${Y}_{{\rm{e}}}$ over the range [0, 1]. The index ${\mathtt{m}}=1,2,3$ allows the user to select three different thermodynamic grid resolutions for the three density ranges $\rho \lt {\rho }_{\mathrm{es}}(1)$, ${\rho }_{\mathrm{es}}(1)\lt \rho \lt {\rho }_{\mathrm{es}}(2)$, and $\rho \gt {\rho }_{\mathrm{es}}(2)$, where ${\rho }_{\mathrm{es}}(1)$ and ρ_es(2) are user selected densities. A particular dependent thermodynamic function, corresponding to the thermodynamic state ($\mathrm{log}\rho ,\mathrm{log}T,{Y}_{{\rm{e}}}$), is computed by linear interpolation from its values at the eight surrounding grid points, which satisfy

$$\begin{eqnarray}\begin{array}{ll}\mathrm{log}{\rho }_{i}\lt \mathrm{log}\rho \leqslant \mathrm{log}{\rho }_{i+1},\quad \mathrm{log}{T}_{j}\lt \mathrm{log}T\leqslant \mathrm{log}{T}_{j+1}, & \ {Y}_{{\rm{e}},k+1}\lt {Y}_{{\rm{e}}}\leqslant {Y}_{{\rm{e}},k},\end{array}\end{eqnarray} \tag{ 5 }$$

where the ρ_i, T_j, and ${Y}_{{\rm{e}},k}$ are the values of ρ, T, and ${Y}_{{\rm{e}}}$ at the grid points. Figure 6 shows a cell of the thermodynamic grid within which the thermodynamic state, ($\mathrm{log}\rho ,\mathrm{log}T,{Y}_{{\rm{e}}}$), is located. We will refer to the eight grid points surrounding a given mass zone as the "surrounding grid points," and the cell itself simply as the "EoS cell." We emphasize that not all grid points of the thermodynamic grid have thermodynamic quantities evaluated and stored there, but only those grid points surrounding the thermodynamic states of mass zones, with a cell of grid points tied to each zone. Initially, the thermodynamic state of each mass zone gets a suite of thermodynamic quantities computed and stored at the eight surrounding grid points. During a simulation, when the changing thermodynamic state of a mass zone causes the state to enter a different EoS cell, the needed thermodynamic quantities are in turn computed and stored on the grid points of that cell. Thus, thermodynamic quantities (and neutrino opacities) are computed on an "as needed" basis, keeping the thermodynamic state of each mass zone surrounded by the needed thermodynamic quantities on the eight nearest ($\mathrm{log}\rho ,\mathrm{log}T,{Y}_{{\rm{e}}}$)-grid points. Lastly, to avoid involving an excessive number of quantities in internode communication when transposing from one set of rays (radial, angular, or azimuthal) to another, the EoS grids along these rays are maintained independently.

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A total of 14 dependent EoS variables comprise the thermodynamic vector that is computed and stored at each of the grid points of a cell surrounding the thermodynamic state of a mass zone. These quantities are the pressure, p; specific internal energy, e; specific entropy, s; neutron chemical potential, μ_n; proton chemical potential, μ_p; electron chemical potential, μ_e; neutron mass fraction, X_n; proton mass fraction, X_p; representative heavy nucleus mass fraction X_H (nuclei with mass numbers greater than helium), along with the mass number, A, charge number Z, and mean binding energy per particle b_A of the representative heavy nucleus; the adiabatic exponent ${{\rm{\Gamma }}}_{s}={\left(\partial p/\partial \rho \right)}_{s,{Y}_{{\rm{e}}}}$, and the specific internal energy, e_int, with the particle rest masses and arbitrary constants subtracted out. The helium mass fraction, X_α, is not stored but is computed as ${X}_{\alpha }=1-{X}_{{\rm{n}}}-{X}_{{\rm{p}}}-{X}_{{\rm{a}}}$. The specific internal energy, e_int, is used to compute the quantity ${{\rm{\Gamma }}}_{e}=p/{e}_{\mathrm{int}}\rho +1$, utilized by the Riemann solver to prevent unphysical ${{\rm{\Gamma }}}_{e}\lt 4/3$, which can cause post-shock oscillations in the some of the thermodynamic variables (Buras et al. 2006b). The formula used to interpolate a thermodynamic quantity from its values at the eight surrounding grid points is differentiated (exactly) to obtain partial derivatives of that quantity with respect to either ρ, T, or ${Y}_{{\rm{e}}}$, as needed.

To provide a sense of the accuracy of our EoS interpolation scheme, Figures 7 and 8 show the relative deviation of the pressure, specific internal energy, and the neutron and proton chemical potentials obtained by direct output from our stellar EoS (described below) versus interpolation in the EoS grid, for the grid resolution listed in the figures. The ρ, s, and ${Y}_{{\rm{e}}}$ profiles used for generating Figures 7 and 8 are representative of models near bounce but before the formation of the shock, and many tens of ms after bounce during shock stagnation. We will refer to these profiles as "Near Bounce" (Figure 7) and "Shock Stagnation" (Figure 8). It is clear from the figures that increasing the density resolution from 10 to 20 grid points per decade in density does not decrease the relative deviation of these thermodynamic quantities except for densities above nuclear saturation. (The black and red curves lie on top of each other below the saturation density.) Increasing the temperature resolution from 20 to 40 points per decade in temperature (orange curve) reduces the relative deviation for all of the graphed quantities throughout most of the density range displayed. Increasing the electron fraction resolution from 50 to 100 over the range [0, 1] in ${Y}_{{\rm{e}}}$ decreases the relative deviation for quantities (namely, abundances and chemical potentials) in regimes (low entropy) where there is a strong dependence on ${Y}_{{\rm{e}}}$—e.g., where there is partial dissociation. The B-series models were performed with the EoS grid resolution of (d-, t-, ygrid) = 20, 20, 50, which, except for a few exceptions described below, typically obtains values of interpolated thermodynamic quantities within a percent or so of the values obtained directly from the EoS (in most cases less than a percent). The D-series models are being performed with the higher grid resolution of (d-, t-, ygrid) = 20, 40, 100, which typically gives a relative deviation about five times smaller.

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A few features of the graphs deserve comment. One feature is the slight kink in the temperature at ρ = 10¹¹ ${{\rm{g\; cm}}}^{-3}$ in the "Near Bounce" profile (Figure 7(a)), and is is due to the LS EoS—C EoS (see Section 3.2) transition at that density. Another is the substantial relative error in the neutron chemical potential at a density of 2 × 10¹⁴ ${{\rm{g\; cm}}}^{-3}$ for the "Near Bounce" profile, and at a density of 2.5 × 10¹⁴ ${{\rm{g\; cm}}}^{-3}$ for the "Shock Stagnation" profile. These are the densities for the respective profiles at which the neutron chemical potentials pass through zero, so any slight deviation in the interpolated versus the directly obtained values for these quantities will be amplified by their small absolute values when computing their relative deviations. The region where the neutron chemical potentials change sign is shown in Figure 9 for the "Near Bounce" profile but is representative of the "Shock Stagnation" profile as well. The spikes in the relative deviation of all quantities at 1.3 × 10¹⁴ ${{\rm{g\; cm}}}^{-3}$ for the "Near Bounce" profile deserve mention, as well. These are caused by the nuclei–nuclear matter phase transition at that density, and the abrupt change in composition there. Table interpolation smooths this transition across the width of the density grid, while direct calls to the EoS see this transition as a discontinuity. These spikes do not appear in the "Shock Stagnation" profile because the higher entropy results in matter being completely dissociated at the above density, leading to a smoother transition across the nuclei–nuclear matter phase transition.

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For CCSN simulations, the pressure, specific internal energy, and specific entropy are taken as the sum of contributions from different species, namely,

$$\begin{eqnarray}\begin{array}{rclrclrcl}p & = & {p}_{\mathrm{ion}}+{p}_{{{\rm{e}}}^{-}+{{\rm{e}}}^{+}}+{p}_{\mathrm{rad}},\, & e & = & {e}_{\mathrm{ion}}+{e}_{{{\rm{e}}}^{-}+{{\rm{e}}}^{+}}+{e}_{\mathrm{rad}},\, & s & = & {s}_{\mathrm{ion}}+{s}_{{{\rm{e}}}^{-}+{{\rm{e}}}^{+}}+{s}_{\mathrm{rad}},\end{array}\end{eqnarray} \tag{ 6 }$$

where the subscripts "ion," "e⁻ + e⁺," and "rad" denote contributions from nuclei, electrons and positrons, and photons, respectively. For the B-, C-, and D-series runs, Chimera employs, for densities above 10¹¹ ${{\rm{g\; cm}}}^{-3}$, the K = 220 MeV incompressibility version of the Lattimer & Swesty (1991) (LS) EoS for the ion and photon components. (The retroactively named A-series used the K = 180 MeV version of LS EoS.) The LS EoS utilizes a compressible liquid drop model for nuclei modeled after the Lamb et al. (1978) (LLPR) formalism, and considers an ion composition of free neutrons and protons, helium, and a representative heavy nucleus. For matter in NSE at densities below 10¹¹ ${{\rm{g\; cm}}}^{-3}$ with ${Y}_{{\rm{e}}}\lt {}^{26}{/}_{56}$, the Cooperstein (1985) (C) EoS is used. The C EoS does not treat the high-density parameters of the liquid drop model (nuclear incompressibility modulus, surface energy, symmetry energy) as consistently as the LS EoS, but it computes the mass fraction of helium more accurately than the LS EoS in the regime below 10¹¹ ${{\rm{g\; cm}}}^{-3}$ where it is being employed. Advection of material across this EoS boundary requires consistent tracking of the specific internal energy. (See Section 4.4.3 for details.) To improve the fidelity of the composition of matter that may eventually become part of the ejecta, in regions of NSE where ${Y}_{{\rm{e}}}\geqslant {}^{26}{/}_{56}$ (the value of Z/A for ${}^{56}\mathrm{Fe}$) the NSE calculation in C EoS has been upgraded to a 17-species representation of the composition, including free neutrons, free protons, the 14 even-Z and even-A nuclei between ${}^{4}\mathrm{He}$ and ${}^{60}\mathrm{Zn}$ plus ${}^{56}\mathrm{Fe}$. This NSE calculation functions similarly to others in the literature (e.g., Clifford & Tayler 1965; Hartmann et al. 1985) but is limited by the small isotope set.

In zones where the timescale required to reach NSE is larger than other physical timescales (e.g., those associated with the evolution of stellar core fluid), the nuclear composition is evolved using the XNet thermonuclear reaction network code. In these regions, the thermodynamic state depends on the isotopic composition, as well as ρ and T, and the electron fraction (${Y}_{{\rm{e}}}$) is calculated from ${Y}_{e}={\sum }_{i}{Z}_{i}{Y}_{i}$, where Z_i is the proton number of an isotope and Y_i is the molar abundance of that isotope.

The initial value problem presented by a nuclear reaction network for an isolated region (individual zone) can, in principle, be solved by a wide range of methods discussed in the literature. However the physical nature of the problem, reflected in the wide range of reaction timescales, renders these numerical systems stiff. The challenges of solving such stiff astrophysical systems are detailed in a number of review articles on the subject (see, e.g., Hix & Meyer 2006; Travaglio & Hix 2013). In the A-, B- and C-series of Chimera models, XNet utilizes the fully implicit Backward-Euler method, introduced to nuclear astrophysics by Arnett & Truran (1969). Data for these reactions is drawn from the REACLIB compilation (Rauscher & Thielemann 2000). Unfortunately, the A- and B-series Chimera models neglect screening of nuclear reactions. The nuclear state is updated for each non-NSE zone in each time step. As needed in each zone, the network is sub-cycled until the hydrodynamic time step is reached.

Several pre-built networks available in Chimera are shown in Table 1. The A- and B-series models all utilize the simple 14-species $\alpha$-network alpha. The active nuclear material evolved in XNet excludes free protons, free neutrons, and an auxiliary heavy nucleus that are advected with the nuclear composition. In the C-series models, we switched to the alpnp network that adds protons and neutrons to the network, though the free nucleons are effectively inert, as their are no reactions included that connect them to other species. The properties (mass and charge number, binding energy, and mass fraction) of the auxiliary heavy nucleus are taken initially from the part of the composition of the progenitor that cannot be mapped onto the network species. For material that has come out of NSE, the properties are taken from the sum of nuclei not included in the network composition vector. For networks alpha and alpnp, this consists of ${}^{56}\mathrm{Fe}$ for ${Y}_{{\rm{e}}}\geqslant {}^{26}{/}_{56}$, or the representative heavy nucleus from the nuclear EoS for ${Y}_{{\rm{e}}}\leqslant {}^{26}{/}_{56}$. For the various D-series models underway, the base network has been updated to anp56, which adds ${}^{56}\mathrm{Fe}$ as an additional unconnected, inert species and permits the network to map directly to the 17-species NSE used by the extended C-EoS and also reduces the mass fraction traced by the auxiliary heavy nucleus. These modifications to the network infrastructure primarily serve the development of even larger networks (sn150 and sn160) for CCSN simulations. The D-series includes 2D and 3D simulations utilizing the sn160 network.

Table 1.  Available Nuclear Networks

Network	Species
alpha	${}^{4}\mathrm{He}$, ${}^{12}{\rm{C}}$, ${}^{16}{\rm{O}}$, ${}^{20}\mathrm{Ne}$, ${}^{24}\mathrm{Mg}$, ${}^{28}\mathrm{Si}$, ${}^{32}{\rm{S}}$, ${}^{36}\mathrm{Ar}$, ${}^{40}\mathrm{Ca}$, ${}^{44}\mathrm{Ti}$, ${}^{48}\mathrm{Cr}$, ${}^{52}\mathrm{Fe}$, ${}^{56}\mathrm{Ni}$, ${}^{60}\mathrm{Zn}$
alpnp	${}^{}{\rm{n}}$ a, ${}^{1}{\rm{H}}$ a, ${}^{4}\mathrm{He}$, ${}^{12}{\rm{C}}$, ${}^{16}{\rm{O}}$, ${}^{20}\mathrm{Ne}$, ${}^{24}\mathrm{Mg}$, ${}^{28}\mathrm{Si}$, ${}^{32}{\rm{S}}$, ${}^{36}\mathrm{Ar}$, ${}^{40}\mathrm{Ca}$, ${}^{44}\mathrm{Ti}$, ${}^{48}\mathrm{Cr}$, ${}^{52}\mathrm{Fe}$, ${}^{56}\mathrm{Ni}$, ${}^{60}\mathrm{Zn}$
anp56	${}^{}{\rm{n}}$ a, ${}^{1}{\rm{H}}$ a, ${}^{4}\mathrm{He}$, ${}^{12}{\rm{C}}$, ${}^{16}{\rm{O}}$, ${}^{20}\mathrm{Ne}$, ${}^{24}\mathrm{Mg}$, ${}^{28}\mathrm{Si}$, ${}^{32}{\rm{S}}$, ${}^{36}\mathrm{Ar}$, ${}^{40}\mathrm{Ca}$, ${}^{44}\mathrm{Ti}$, ${}^{48}\mathrm{Cr}$, ${}^{52}\mathrm{Fe}$, ${}^{56}\mathrm{Fe}$ a, ${}^{56}\mathrm{Ni}$, ${}^{60}\mathrm{Zn}$
	${}^{}{\rm{n}}$, ${}^{1-2}{\rm{H}}$, ${}^{3-4}\mathrm{He}$, ${}^{6-7}\mathrm{Li}$, ${}^{\mathrm{7,9}}\mathrm{Be}$, ${}^{\mathrm{8,10,11}}{\rm{B}}$, ${}^{12-14}{\rm{C}}$, ${}^{13-15}{\rm{N}}$, ${}^{14-18}{\rm{O}}$, ${}^{17-19}{\rm{F}}$, ${}^{18-22}\mathrm{Ne}$, ${}^{21-23}\mathrm{Na}$,
sn150	${}^{23-26}\mathrm{Mg}$, ${}^{24-27}\mathrm{Al}$, ${}^{28-32}\mathrm{Si}$, ${}^{29-33}{\rm{P}}$, ${}^{32-36}{\rm{S}}$, ${}^{33-37}\mathrm{Cl}$, ${}^{36-40}\mathrm{Ar}$, ${}^{37-41}{\rm{K}}$, ${}^{40-48}\mathrm{Ca}$, ${}^{43-49}\mathrm{Sc}$,
	${}^{44-50}\mathrm{Ti}$, ${}^{46-51}{\rm{V}}$, ${}^{48-54}\mathrm{Cr}$, ${}^{50-55}\mathrm{Mn}$, ${}^{52-58}\mathrm{Fe}$, ${}^{53-59}\mathrm{Co}$, ${}^{56-62}\mathrm{Ni}$, ${}^{57-63}\mathrm{Cu}$, ${}^{59-66}\mathrm{Zn}$
	${}^{}{\rm{n}}$, ${}^{1-2}{\rm{H}}$, ${}^{3-4}\mathrm{He}$, ${}^{6-7}\mathrm{Li}$, ${}^{\mathrm{7,9}}\mathrm{Be}$, ${}^{\mathrm{8,10,11}}{\rm{B}}$, ${}^{12-14}{\rm{C}}$, ${}^{13-15}{\rm{N}}$, ${}^{14-18}{\rm{O}}$, ${}^{17-19}{\rm{F}}$, ${}^{18-22}\mathrm{Ne}$, ${}^{21-23}\mathrm{Na}$,
sn160	${}^{23-26}\mathrm{Mg}$, ${}^{25-27}\mathrm{Al}$, ${}^{28-32}\mathrm{Si}$, ${}^{29-33}{\rm{P}}$, ${}^{32-36}{\rm{S}}$, ${}^{33-37}\mathrm{Cl}$, ${}^{36-40}\mathrm{Ar}$, ${}^{37-41}{\rm{K}}$, ${}^{40-48}\mathrm{Ca}$, ${}^{43-49}\mathrm{Sc}$, ${}^{44-51}\mathrm{Ti}$,
	${}^{46-52}{\rm{V}}$, ${}^{48-54}\mathrm{Cr}$, ${}^{50-55}\mathrm{Mn}$, ${}^{52-58}\mathrm{Fe}$, ${}^{53-59}\mathrm{Co}$, ${}^{56-64}\mathrm{Ni}$, ${}^{57-65}\mathrm{Cu}$, ${}^{59-66}\mathrm{Zn}$, ${}^{62-64}\mathrm{Ga}$, ${}^{63-64}\mathrm{Ge}$

Note.

^aInert species, which are advected but not reactive.

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When not in NSE, the thermodynamic quantities for the EoS cell are computed assuming the same composition of nuclei on all eight vertices. This gives rise to an apparent inconsistency, however, as the electron fraction, ${Y}_{{\rm{e}}}$, at some or all of the EoS grid points will not correspond to the ${Y}_{{\rm{e}}}$ of the composition of nuclei. Furthermore, the electron–positron contribution to the thermodynamic vector at each EoS grid point is computed from the values of ρ, T, and ${Y}_{{\rm{e}}}$ at that grid point, the result being that the ${Y}_{{\rm{e}}}$ of the electron–positron gas is not consistent with the ${Y}_{{\rm{e}}}$ of the composition of nuclei at a grid point. This procedure, however, allows us to take finite derivatives with respect to the electron fraction, albeit with some approximation, but accurate enough to stabilize the neutrino transport in the non-NSE regions. At the same time, when the thermodynamic state of a mass zone is interpolated from the EoS cell, the contribution of the ions will be based on a ${Y}_{{\rm{e}}}$ common to all EoS cell points, while the contributions of the electron–positron gas will be interpolated to the same ${Y}_{{\rm{e}}}$, and the two contributions will reflect the same value of ${Y}_{{\rm{e}}}$. The ions used to compute the thermodynamic properties are those in the network, the heavy nucleus advected with the active composition, and, in the case of the original alpha network used in Series A and B, the free nucleon mass fractions whose values are also stored with the thermodynamic vector.

Chimera's treatment of the transition of matter into NSE is comparable to that used in other CCSN codes of similar capability (Buras et al. 2006a; Müller et al. 2012; Nakamura et al. 2014; Skinner et al. 2019). The transition condition is motivated by the temperatures and densities at which complete silicon burning would occur within the current global time step. For temperatures above this threshold, the use of the nuclear network is superfluous, as the network will achieve NSE every time step. For transitioning into NSE, there will be a slight change in the nuclear binding energy, due to the change of the representative nucleus, the ensemble of nuclei, etc., and the extent of this change is one metric for gauging the accuracy of our assumption of NSE. In order to maintain hydrodynamic stability across this transition, we adjust temperature to maintain constant pressure for a given density and electron fraction. This may result in a small change in the specific internal energy (including binding energy), but the dynamical impact that results from the transition between inconsistent thermodynamic states is minimized. The advection of material across an NSE/non-NSE interface in either direction, as well as the transition into NSE and freeze-out from NSE of entire zones, includes the appropriate gain or loss of nuclear binding energy (see Section 4.4.4 for details). In the A-series simulations, the NSE interface was a sphere of fixed radius, while in the B-series simulations, the NSE boundary was independent for each radial ray. The C- and later series allow multiple NSE/non-NSE interfaces along any coordinate direction.

To determine whether a zone is in NSE and may, therefore, be omitted from nuclear burning, Chimera applies an empirically determined linear relationship between the NSE transition temperature, ${T}_{\mathrm{NSE}}$, and the density:

$$\begin{eqnarray}{T}_{\mathrm{NSE}}(\rho )=\left\{\begin{array}{ll}{C}_{1}\rho +{C}_{2} & \mathrm{if}\ \rho \lt 2\,\times {10}^{8}\,{{\rm{g\; cm}}}^{-3};\\ 6.5\,\times {10}^{9}\,{\rm{K}} & \mathrm{otherwise},\end{array}\right.\end{eqnarray} \tag{ 7 }$$

where C₁ ≡ 5.333 ${\rm{K}}\,{{\rm{g}}}^{-1}\,{\mathrm{cm}}^{3}$ and ${C}_{2}\equiv 5.433\times {10}^{9}\,{\rm{K}}$. At the beginning of a global timestep, any non-NSE zone for which $T\geqslant {T}_{\mathrm{NSE}}$ is transitioned to NSE. A zone which is in NSE at the beginning of a timestep will be transitioned out of NSE if $T\lt ({T}_{\mathrm{NSE}}-2\times {10}^{8}\,{\rm{K}})$ and if the representative heavy nucleus, split into ${}^{56}\mathrm{Ni}$ and ${}^{56}\mathrm{Fe}$, will result in less than half of the mass fraction being ${}^{56}\mathrm{Fe}$. If this latter condition is not met, NSE is maintained as long as is practical, until the condition is met or until $T\lt 4.9\times {10}^{9}\,{\rm{K}}$, as the NSE representation of neutron-rich composition is better than that allowed by this limited network.

For simplicity, the transition out of NSE occurs when the temperature drops below this NSE condition (Equation (7) for Chimera). However, for the rapidly changing conditions in expanding CCSN matter, the assumption of NSE has been shown to break down when the temperature falls below 6 GK (Meyer et al. 1998). For transitioning out of NSE using the alpha network (but before evolving the network), the nuclear binding energy does not change. The NSE composition is evaluated using some analytically calculated state, but the temperature is then adjusted so that, for a given density and electron fraction, we will get the same specific internal energy (including binding energy) using a local EoS cube interpolation.

The computation of the electron–positron component of the EoS is divided into five regimes (Figure 10) as described below. The first major division is based on whether the electrons are relativistic or nonrelativistic. The electron–positron gas is regarded as nonrelativistic if

$$\begin{eqnarray}&&{e}_{\mathrm{nr},\mathrm{Fermi}}\lt 0.01{m}_{{\rm{e}}}{c}^{2}\qquad {kT}\lt 0.01\,{m}_{{\rm{e}}}{c}^{2},\end{eqnarray} \tag{ 8 }$$

where the nonrelativistic electron Fermi energy, ${e}_{\mathrm{nr}\mathrm{Fermi}}$, is given by

$$\begin{eqnarray}&&{e}_{\mathrm{nr},\mathrm{Fermi}}=\displaystyle \frac{{\left({\hslash }c\right)}^{2}}{2{m}_{{\rm{e}}}{c}^{2}}{\left(3{\pi }^{2}\right)}^{2/3}{n}_{{\rm{e}}}^{2/3},\quad {n}_{{\rm{e}}}=\displaystyle \frac{\rho {Y}_{{\rm{e}}}}{{m}_{{\rm{B}}}},\end{eqnarray} \tag{ 9 }$$

where ${n}_{{\rm{e}}}={n}_{{{\rm{e}}}^{-}}-{n}_{{{\rm{e}}}^{+}}$ is the net electron number density, ${n}_{{{\rm{e}}}^{-}}$ is the electron density, and ${n}_{{{\rm{e}}}^{+}}$ is the positron density. Otherwise, the electron–positron gas is treated as relativistic.

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If the electron–positron gas is relativistic, there are three regimes where different approximations are used: high temperature, very degenerate, and intermediate. In the intermediate regime, the Fermi integrals for the electron–positron thermodynamic functions are integrated numerically. Since this is the most computationally intensive procedure, it is first ascertained whether the thermodynamic state can be considered to be in the high-temperature or the very degenerate regime. If the thermodynamic state is found to be in neither of those regimes, the calculation of the thermodynamic functions begins with net electron density, given by

$$\begin{eqnarray}\begin{array}{rcl}{n}_{{\rm{e}}} & = & \displaystyle \frac{1}{{\pi }^{2}{{\hslash }}^{3}}\left[{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{p}^{2}{dp}}{\exp \left[(\sqrt{{p}^{2}{c}^{2}+{m}_{{\rm{e}}}^{2}{c}^{4}}-{\mu }_{{\rm{e}}})/{kT}\right]+1}\right.-\left.\,{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{p}^{2}{dp}}{\exp \left[(\sqrt{{p}^{2}{c}^{2}+{m}_{{\rm{e}}}^{2}{c}^{4}}+{\mu }_{{\rm{e}}})/{kT}\right]+1}\right]\\ & = & \displaystyle \frac{{k}^{3}{T}^{3}}{{\pi }^{2}{{\hslash }}^{3}}\left[{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{\left(x+\beta \right){dx}\sqrt{x(x+2\beta )}}{{e}^{x+\beta -{\eta }_{{\rm{e}}}}+1}\right.-\,\left.{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{\left(x+\beta \right){dx}\sqrt{x(x+2\beta )}}{{e}^{x+\beta +{\eta }_{{\rm{e}}}}+1}\right],\end{array}\end{eqnarray} \tag{ 10 }$$

where in the last expression $\beta ={m}_{{\rm{e}}}{c}^{2}/{kT}$, ${\eta }_{{\rm{e}}}={\mu }_{{\rm{e}}}/{kT}$, and the following substitutions have been made: $y=p/{m}_{{\rm{e}}}c$, ${z}^{2}={y}^{2}+1$, and $x=\beta (z-1)$. To obtain ${\eta }_{{\rm{e}}}$, the right-hand side of Equation (10) is integrated numerically by means of a 48-point Gauss–Laguerre scheme using a guess for η_e, and then iterated for the η_e until the right-hand side equals n_e to within one part in 10⁶. Once η_e is obtained, ${p}_{{\rm{e}}+{\rm{p}}}$ and ${e}_{{\rm{e}}+{\rm{p}}}$, the latter of which includes the electron–positron rest mass energy, are obtained by 48-point Gauss–Laguerre numerical integration of the following Fermi integrals

$$\begin{eqnarray}\begin{array}{rclrcl}{p}_{{\rm{e}}+{\rm{p}}} & = & \displaystyle \frac{{\hslash }c}{3{\pi }^{2}}{\left(\displaystyle \frac{{kT}}{{\hslash }c}\right)}^{4}{\displaystyle \int }_{0}^{\infty }\left[{dx}\displaystyle \frac{{\left({x}^{2}+2\beta x\right)}^{3/2}}{{e}^{x+\beta -{\eta }_{{\rm{e}}}}+1}\right. & & & +\left.\,{dx}\displaystyle \frac{{\left({x}^{2}+2\beta x\right)}^{3/2}}{{e}^{x+\beta +{\eta }_{{\rm{e}}}}+1}\right],\end{array}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}\begin{array}{rclrcl}{e}_{{\rm{e}}+{\rm{p}}} & = & \displaystyle \frac{{\hslash }c}{{\pi }^{2}}{\left(\displaystyle \frac{{kT}}{{\hslash }c}\right)}^{4}{\displaystyle \int }_{0}^{\infty }\left[{dx}\displaystyle \frac{{\left(x+\beta \right)}^{2}\sqrt{{x}^{2}+2\beta x}}{{e}^{x+\beta -{\eta }_{{\rm{e}}}}+1}\right. & & & +\,\left.{dx}\displaystyle \frac{{\left(x+\beta \right)}^{2}\sqrt{{x}^{2}+2\beta x}}{{e}^{x+\beta +{\eta }_{{\rm{e}}}}+1}\right].\end{array}\end{eqnarray} \tag{ 12 }$$

The entropy is obtained from the pressure, specific internal energy, and chemical potential from the thermodynamic relation

$$\begin{eqnarray}&&{s}_{{\rm{e}}+{\rm{p}}}=\displaystyle \frac{1}{{kT}}\left[\displaystyle \frac{{e}_{{\rm{e}}+{\rm{p}}}}{{n}_{{\rm{B}}}}+\displaystyle \frac{{p}_{{\rm{e}}+{\rm{p}}}}{{n}_{{\rm{B}}}}-{\mu }_{{\rm{e}}}{Y}_{{\rm{e}}}\right].\end{eqnarray} \tag{ 13 }$$

To determine whether the high-temperature approximation may be applied, Equation (10) is rewritten with the substitution z = x + β to get

$$\begin{eqnarray}&&{n}_{{\rm{e}}}=\displaystyle \frac{1}{{\pi }^{2}}{\left(\displaystyle \frac{{kT}}{{\hslash }c}\right)}^{3}\left[{\displaystyle \int }_{\beta }^{\infty }\displaystyle \frac{{zdz}\sqrt{{z}^{2}-{\beta }^{2}}}{{e}^{z-{\eta }_{{\rm{e}}}}+1}-{\displaystyle \int }_{\beta }^{\infty }\displaystyle \frac{{zdz}\sqrt{{z}^{2}-{\beta }^{2}}}{{e}^{z+{\eta }_{{\rm{e}}}}+1}\right].\end{eqnarray} \tag{ 14 }$$

The high-temperature approximation consists of setting β = 0 in Equation (14) and performing the integrations analytically

$$\begin{eqnarray}\begin{array}{rcl}{n}_{{\rm{e}}} & \simeq & \displaystyle \frac{1}{{\pi }^{2}}{\left(\displaystyle \frac{{kT}}{{\hslash }c}\right)}^{3}\left[{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{x}^{2}{dx}}{{e}^{x-{\eta }_{{\rm{e}}}}+1}-{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{x}^{2}{dx}}{{e}^{x+{\eta }_{{\rm{e}}}}+1}\right]\\ & = & \displaystyle \frac{1}{{\pi }^{2}}{\left(\displaystyle \frac{{kT}}{{\hslash }c}\right)}^{3}\left[{F}_{2}({\eta }_{{\rm{e}}})-{F}_{2}(-{\eta }_{{\rm{e}}})\right]\\ & = & \displaystyle \frac{1}{{\pi }^{2}}{\left(\displaystyle \frac{{kT}}{{\hslash }c}\right)}^{3}\left[\displaystyle \frac{1}{3}{\eta }_{{\rm{e}}}\left({\pi }^{2}+{\eta }_{{\rm{e}}}^{2}\right)\right],\end{array}\end{eqnarray} \tag{ 15 }$$

with the solution for η_e

$$\begin{eqnarray}&&{\eta }_{{\rm{e}}}=\sqrt[3]{\displaystyle \frac{3\chi }{2}+\sqrt{\displaystyle \frac{9{\chi }^{2}}{4}+\displaystyle \frac{{\pi }^{6}}{27}}}+\sqrt[3]{\displaystyle \frac{3\chi }{2}-\sqrt{\displaystyle \frac{9{\chi }^{2}}{4}+\displaystyle \frac{{\pi }^{6}}{27}}},\end{eqnarray} \tag{ 16 }$$

where $\chi ={\pi }^{2}{\left({\hslash }c/{kT}\right)}^{3}{n}_{{\rm{e}}}$. The high-temperature approximation is then applied if β < 2/3, or if β < 2 and η_e given by Equation (16) satisfies η_e > 2β. The high-temperature approximation is applied if values of ρ and T lie above and to the right of the solid red line in Figure 10. The solid red horizontal line at the left is given by the first condition above, the rest of the red line is given by the second condition. The solid green line terminating in the curved red segment is given by the condition η_e > 2β. Once η_e is obtained, analytic expressions for ${p}_{{\rm{e}}+{\rm{p}}}$ and ${e}_{{\rm{e}}+{\rm{p}}}$ in the high-temperature approximation are given by

$$\begin{eqnarray}\begin{array}{rcl}{p}_{{\rm{e}}+{\rm{p}}} & = & \displaystyle \frac{{\hslash }c}{3{\pi }^{2}}{\left(\displaystyle \frac{{kT}}{{\hslash }c}\right)}^{4}\,\left[{F}_{3}({\eta }_{{\rm{e}}})+{F}_{3}(-{\eta }_{{\rm{e}}})-\displaystyle \frac{3}{2}{\beta }^{2}\left[{F}_{1}({\eta }_{{\rm{e}}})+{F}_{1}(-{\eta }_{{\rm{e}}})\right]\right]\\ & = & \,\displaystyle \frac{{\hslash }c}{3{\pi }^{2}}{\left(\displaystyle \frac{{kT}}{{\hslash }c}\right)}^{4}\,\left[\displaystyle \frac{7{\pi }^{4}}{60}+\displaystyle \frac{1}{2}{\eta }_{{\rm{e}}}^{2}\left({\pi }^{2}+\displaystyle \frac{1}{2}{\eta }_{{\rm{e}}}^{2}\right)-\displaystyle \frac{3}{2}\displaystyle \frac{{\beta }^{2}}{6}\left({\pi }^{2}+3{\eta }_{{\rm{e}}}^{2}\right)\right],\end{array}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}\begin{array}{rcl}{e}_{{\rm{e}}+{\rm{p}}} & = & \displaystyle \frac{{\hslash }c}{{\pi }^{2}}{\left(\displaystyle \frac{{kT}}{{\hslash }c}\right)}^{4}\,\left[{F}_{3}({\eta }_{{\rm{e}}})+{F}_{3}(-{\eta }_{{\rm{e}}})-\displaystyle \frac{1}{2}{\beta }^{2}\left[{F}_{1}({\eta }_{{\rm{e}}})+{F}_{1}(-{\eta }_{{\rm{e}}})\right]\right],\end{array}\end{eqnarray} \tag{ 18 }$$

so that

$$\begin{eqnarray}\begin{array}{rcl}{e}_{{\rm{e}}+{\rm{p}}}+{p}_{{\rm{e}}+{\rm{p}}} & = & \displaystyle \frac{{\hslash }c}{3{\pi }^{2}}{\left(\displaystyle \frac{{kT}}{{\hslash }c}\right)}^{4}\left\{4\left({F}_{3}({\eta }_{{\rm{e}}})+{F}_{3}(-{\eta }_{{\rm{e}}}\right)\right.-4\times \displaystyle \frac{3}{2}{\beta }^{2}\left[{F}_{1}({\eta }_{{\rm{e}}})+{F}_{1}(-{\eta }_{{\rm{e}}})\right]\\ & & \left.+2\times \displaystyle \frac{3}{2}{\beta }^{2}\left[{F}_{1}({\eta }_{{\rm{e}}})+{F}_{1}(-{\eta }_{{\rm{e}}})\right]\right\}\\ & = & 4{p}_{{\rm{e}}+{\rm{p}}}+\displaystyle \frac{{\hslash }c}{{\pi }^{2}}{\left(\displaystyle \frac{{kT}}{{\hslash }c}\right)}^{4}\displaystyle \frac{{\beta }^{2}}{6}\left({\pi }^{2}+3{\eta }_{{\rm{e}}}^{2}\right),\end{array}\end{eqnarray} \tag{ 19 }$$

where the F's are the usual Fermi integrals. Substituting Equation (19) into Equation (13) for the entropy gives

$$\begin{eqnarray}&&{s}_{{\rm{e}}+{\rm{p}}}=\displaystyle \frac{1}{{{kTn}}_{{\rm{B}}}}\left[4{p}_{{\rm{e}}+{\rm{p}}}+\displaystyle \frac{{\hslash }c}{{\pi }^{2}}{\left(\displaystyle \frac{{kT}}{{\hslash }c}\right)}^{4}\displaystyle \frac{{\beta }^{2}}{6}\left({\pi }^{2}+3{\eta }_{{\rm{e}}}^{2}\right)\right]-{\eta }_{{\rm{e}}}{Y}_{{\rm{e}}}.\end{eqnarray} \tag{ 20 }$$

After computing the pressure from Equation (17), the entropy is computed from Equation (20), and then the specific internal energy is computed from the thermodynamic relation

$$\begin{eqnarray}&&{e}_{{\rm{e}}+{\rm{p}}}={n}_{{\rm{B}}}{kT}\left({s}_{{\rm{e}}+{\rm{p}}}+{\eta }_{{\rm{e}}}{Y}_{{\rm{e}}}\right)-{p}_{{\rm{e}}+{\rm{p}}}.\end{eqnarray} \tag{ 21 }$$

The thermodynamic state is considered to be in the very degenerate regime if, as computed by Equation (16) or determined by iterating Equation (10), η_e > 35. In this case, the relativistic Sommerfeld approximation is used for the thermodynamic functions, starting with

$$\begin{eqnarray}\begin{array}{rcl}{n}_{{{\rm{e}}}^{-}} & \simeq & \displaystyle \frac{1}{3{\pi }^{2}}{\left(\displaystyle \frac{{m}_{{\rm{e}}}{c}^{2}}{{\hslash }c}\right)}^{3}\left\{{x}_{\eta }^{3}+{\left(\displaystyle \frac{{kT}}{{m}_{{\rm{e}}}{c}^{2}}\right)}^{2}\displaystyle \frac{{\pi }^{2}}{2}\right.\left.\,\left[\displaystyle \frac{1}{{x}_{\eta }}\left(2{x}_{\eta }^{2}+1\right)\right]+{\left(\displaystyle \frac{{kT}}{{m}_{{\rm{e}}}{c}^{2}}\right)}^{4}\displaystyle \frac{7{\pi }^{4}}{40}{x}_{\eta }^{-5}\right\},\end{array}\end{eqnarray} \tag{ 22 }$$

which is iterated on ${x}_{\eta }={pc}/{m}_{{\rm{e}}}{c}^{2}$, where p is the electron momentum, until the right-hand side is equal to ${n}_{{{\rm{e}}}^{-}}$ to one part in 10⁶. Once x_η is determined, the other thermodynamic functions are computed as follows:

$$\begin{eqnarray}\begin{array}{rcl}{p}_{{\rm{e}}} & = & \displaystyle \frac{{\left({m}_{{\rm{e}}}{c}^{2}\right)}^{4}}{{\left({\hslash }c\right)}^{3}}\displaystyle \frac{1}{24{\pi }^{2}}\left[\Space{0ex}{4.45ex}{0ex}{x}_{\eta }\left(2{x}_{\eta }^{2}-3\right)\sqrt{{x}_{\eta }^{2}+1}\right.+3\mathrm{log}\left({x}_{\eta }+\sqrt{{x}_{\eta }^{2}+1}\right)\\ & & +\,4{\pi }^{2}{\left(\displaystyle \frac{{kT}}{{m}_{{\rm{e}}}{c}^{2}}\right)}^{2}{x}_{\eta }\sqrt{{x}_{\eta }^{2}+1}+\,\left.{\left(\displaystyle \frac{{kT}}{{m}_{{\rm{e}}}{c}^{2}}\right)}^{4}\displaystyle \frac{7{\pi }^{4}}{15}\displaystyle \frac{\sqrt{{x}_{\eta }^{2}+1}\left(2{x}_{\eta }^{2}-1\right)}{{x}_{\eta }^{3}}\right],\end{array}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}\begin{array}{rcl}{e}_{{\rm{e}}} & = & \displaystyle \frac{{\left({m}_{{\rm{e}}}{c}^{2}\right)}^{4}}{{\left({\hslash }c\right)}^{3}}\displaystyle \frac{1}{24{\pi }^{2}}\left[8{x}_{\eta }^{3}\left(\sqrt{{x}_{\eta }^{2}+1}-1\right)\right.-\,{x}_{\eta }\left(2{x}_{\eta }^{2}-3\right)\sqrt{{x}_{\eta }^{2}+1}-3\mathrm{log}\left({x}_{\eta }+\sqrt{{x}_{\eta }^{2}+1}\right)\\ & & +\,4{\pi }^{2}{\left(\displaystyle \frac{{kT}}{{m}_{{\rm{e}}}{c}^{2}}\right)}^{2}\left(\sqrt{{x}_{\eta }^{2}+1}\left(3{x}_{\eta }^{2}+1\right){/x}_{\eta }\right.-\,\left.\left.\left(2{x}_{\eta }^{2}+1\right){/x}_{\eta }\right)\right]+{n}_{{\rm{B}}}{Y}_{{\rm{e}}}{m}_{{\rm{e}}},\end{array}\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{\mu }_{{\rm{e}}}={m}_{{\rm{e}}}{c}^{2}\sqrt{{x}_{\eta }^{2}+1},\end{eqnarray} \tag{ 25 }$$

and

$$\begin{eqnarray}&&{s}_{{\rm{e}}}=\displaystyle \frac{1}{{kT}}\left(\displaystyle \frac{{e}_{{\rm{e}}}+{p}_{{\rm{e}}}}{{n}_{{\rm{B}}}}-{\mu }_{{\rm{e}}}{Y}_{{\rm{e}}}\right).\end{eqnarray} \tag{ 26 }$$

If the nonrelativistic criteria in Equations (8) are satisfied, then there are two regimes in which different approximations are applied. If ${\epsilon }_{{\rm{F}}}/{kT}\gt 35$, the nonrelativistic Sommerfeld approximation is used, otherwise the nonrelativistic Fermi integrals for the electrons are integrated numerically. In the latter case, the Fermi integral for the electron number density

$$\begin{eqnarray}&&{n}_{{\rm{e}}}=\displaystyle \frac{{2}^{1/2}{\left({m}_{{\rm{e}}}{c}^{2}\right)}^{3/2}{\left({kT}\right)}^{3/2}}{{\pi }^{2}{\left({\hslash }c\right)}^{3}}{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{x}^{1/2}{dx}}{{e}^{x-{\eta }_{{\rm{e}}}}+1}.\end{eqnarray} \tag{ 27 }$$

is integrated by a 48-point Gauss–Laguerre scheme and iterated on η_e until the right and left sides of Equation (27) are equal to one part in 10⁶. Once η_e has been obtained, μ_e is computed by ${\mu }_{{\rm{e}}}={kT}{\eta }_{{\rm{e}}}+{m}_{{\rm{e}}}{c}^{2}$, and the electron pressure, specific internal energy, and entropy are obtained by

$$\begin{eqnarray}&&{e}_{{\rm{e}},\mathrm{kin}}=\displaystyle \frac{3}{2}\displaystyle \frac{{\left(2{m}_{{\rm{e}}}\right)}^{3/2}{\left({kT}\right)}^{5/2}}{3{\pi }^{2}{{\hslash }}^{3}}{\displaystyle \int }_{0}^{\infty }{x}^{3/2}{dx}\displaystyle \frac{1}{{e}^{x-{\eta }_{{\rm{e}}}}+1},\end{eqnarray} \tag{ 28 }$$

and

$$\begin{eqnarray}\begin{array}{rcl}{e}_{{\rm{e}}} & = & {e}_{{\rm{e}},\mathrm{kin}}+\displaystyle \frac{\rho {Y}_{{\rm{e}}}}{{m}_{{\rm{B}}}}{m}_{{\rm{e}}}{c}^{2},\quad {p}_{{\rm{e}}}=\displaystyle \frac{2}{3}{e}_{{\rm{e}},\mathrm{kin}},\\ {s}_{{\rm{e}}} & = & \displaystyle \frac{1}{{kT}}\left(\displaystyle \frac{{e}_{{\rm{e}}}+{p}_{{\rm{e}}}}{{n}_{{\rm{B}}}}-{\mu }_{{\rm{e}}}{Y}_{{\rm{e}}}\right).\end{array}\end{eqnarray} \tag{ 29 }$$

In the case that ${\epsilon }_{{\rm{F}}}/{kT}\gt 35$, the Sommerfeld approximations for the nonrelativistic Fermi expressions are used. Thus, the electron density is given by

$$\begin{eqnarray}&&{n}_{{\rm{e}}}=\displaystyle \frac{{\left(2{m}_{{\rm{e}}}{c}^{2}\right)}^{3/2}{\left({kT}\right)}^{3/2}}{3{\pi }^{2}{\left({\hslash }c\right)}^{3}}{\eta }_{{\rm{e}}}^{3/2}\left[1+\displaystyle \frac{{\pi }^{2}}{8}{\eta }_{{\rm{e}}}^{-2}+\displaystyle \frac{7{\pi }^{4}}{640}{\eta }_{{\rm{e}}}^{-4}\right].\end{eqnarray} \tag{ 30 }$$

Equation (30) is iterated on η_e as described above, μ_e is then computed also as described above, and the electron pressure, specific internal energy, and entropy are obtained by

$$\begin{eqnarray}&&{p}_{{\rm{e}}}=\displaystyle \frac{2{\left(2{m}_{{\rm{e}}}\right)}^{3/2}{\left({kT}\right)}^{5/2}}{15{\pi }^{2}{{\hslash }}^{3}}{\eta }_{{\rm{e}}}^{5/2}\left[1+\displaystyle \frac{5{\pi }^{2}}{8}{\eta }_{{\rm{e}}}^{-2}+\displaystyle \frac{7{\pi }^{4}}{384}{\eta }_{{\rm{e}}}^{-4}\right],\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&{e}_{{\rm{e}}}=\displaystyle \frac{3}{2}{p}_{{\rm{e}}}+\displaystyle \frac{\rho {Y}_{{\rm{e}}}}{{m}_{{\rm{B}}}}{m}_{{\rm{e}}}{c}^{2},\quad {s}_{{\rm{e}}}=\displaystyle \frac{1}{{kT}}\left(\displaystyle \frac{{e}_{{\rm{e}}}+{p}_{{\rm{e}}}}{{n}_{{\rm{B}}}}-{\mu }_{{\rm{e}}}{Y}_{{\rm{e}}}\right).\end{eqnarray} \tag{ 32 }$$

The accuracies of the various approximations are indicated by the relative deviations of the electron–positron pressure and specific internal energy as computed by pairs of these approximations at the boundaries separating their respective regimes of applicability. This is shown in Figure 11, which indicates that relative deviations are at most a few tenths of a percent. The discontinuity between approximation regimes is smoothed by the finite resolution of the EoS table.

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A "2γ," thermodynamically consistent EoS has been developed, simple enough to be completely analytic, yet rich enough to be used for testing the hydrodynamics and transport modules of Chimera. A system of completely degenerate and relativistic free electrons, partially degenerate interacting neutrons, and nondegenerate free protons, is modeled by the free energy

$$\begin{eqnarray}\begin{array}{rcl}F & = & {F}_{1}{\left(\displaystyle \frac{{N}_{e}}{V}\right)}^{{\gamma }_{1}}{\left(\displaystyle \frac{{V}_{0}}{{N}_{0}}\right)}^{{\gamma }_{1}-4/3}V+{F}_{2}{\left(\displaystyle \frac{{N}_{{\rm{n}}}}{V}\right)}^{{\gamma }_{2}}V\\ & & -\,{kT}\left[{N}_{{\rm{n}}}\mathrm{ln}\left({g}_{{\rm{n}}}V\right)+{E}_{\mathrm{coef}}{N}_{{\rm{n}}}\mathrm{ln}\left(\displaystyle \frac{2\pi {m}_{{\rm{B}}}{kT}}{{h}^{2}}\right)-{N}_{{\rm{n}}}\mathrm{ln}{N}_{{\rm{n}}}\right.\left.+{N}_{{\rm{n}}}\Space{0ex}{3.15ex}{0ex}\right]\\ & & -\,{kT}\left[{N}_{{\rm{p}}}\mathrm{ln}\left({g}_{{\rm{p}}}V\right)+{E}_{\mathrm{coef}}{N}_{{\rm{p}}}\mathrm{ln}\left(\displaystyle \frac{2\pi {m}_{{\rm{B}}}{kT}}{{h}^{2}}\right)\right.\left.-{N}_{{\rm{p}}}\mathrm{ln}{N}_{{\rm{p}}}+{N}_{{\rm{p}}}\Space{0ex}{3.15ex}{0ex}\right],\end{array}\end{eqnarray} \tag{ 33 }$$

where m_B is the baryon mass, g_n = g_p = 2 are the statistical weights of the neutrons and protons, k and h have their usual meanings, and γ₁, γ₂, F₁, F₂, E_coef, and ${X}_{{\rm{p}}}\equiv {N}_{{\rm{p}}}$/$(\rho /{m}_{{\rm{B}}})\,=$ ${Y}_{{\rm{e}}}\equiv ({N}_{{{\rm{e}}}^{-}}-{N}_{{{\rm{e}}}^{+}})/(\rho /{m}_{{\rm{B}}})=1-{X}_{{\rm{n}}}\,=$ ${N}_{{\rm{n}}}/(\rho /{m}_{{\rm{B}}})$ are free parameters. The mass fraction of free protons, X_p, is typically taken to be 0.5, and the parameters γ₁ and F₁ are typically chosen to be

$$\begin{eqnarray}&&{\gamma }_{1}=\displaystyle \frac{4}{3},\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{F}_{1}=\displaystyle \frac{3}{4}{\left(\displaystyle \frac{3}{8\pi }\right)}^{1/3}{hc},\end{eqnarray} \tag{ 35 }$$

so that the first term on the right-had side of Equation (33) represents completely degenerate and extremely relativistic free electrons. Given the above expression for the free energy, the pressure is

$$\begin{eqnarray}\begin{array}{rcl}p & = & -{\left(\displaystyle \frac{\partial F}{\partial V}\right)}_{T,{N}_{i}}=\left({\gamma }_{1}-1\right){F}_{1}{\left(\displaystyle \frac{\rho {Y}_{{\rm{e}}}}{{m}_{{\rm{B}}}}\right)}^{{\gamma }_{1}}{\left(\displaystyle \frac{{m}_{{\rm{B}}}}{{\rho }_{0}}\right)}^{{\gamma }_{1}-4/3}+\left({\gamma }_{2}-1\right){F}_{2}{\left(\displaystyle \frac{\rho {X}_{{\rm{n}}}}{{m}_{{\rm{B}}}}\right)}^{{\gamma }_{2}}+\displaystyle \frac{\rho {kT}}{{m}_{{\rm{B}}}}.\end{array}\end{eqnarray} \tag{ 36 }$$

We choose F₂ so that the contributions of the first two terms for p in Equation (36) become equal at ρ_nuc, which, for the case in which Y_p is 0.5, requires that

$$\begin{eqnarray}&&\left({\gamma }_{1}-1\right){F}_{1}{\left(\displaystyle \frac{{\rho }_{\mathrm{nuc}}0.5}{{m}_{{\rm{B}}}}\right)}^{{\gamma }_{1}}{\left(\displaystyle \frac{{m}_{{\rm{B}}}}{{\rho }_{0}}\right)}^{{\gamma }_{1}-4/3}=\left({\gamma }_{2}-1\right){F}_{2}{\left(\displaystyle \frac{{\rho }_{\mathrm{nuc}}0.5}{{m}_{{\rm{B}}}}\right)}^{{\gamma }_{2}}\end{eqnarray} \tag{ 37 }$$

or

$$\begin{eqnarray}&&{F}_{2}=\displaystyle \frac{{\gamma }_{1}-1}{{\gamma }_{2}-1}{F}_{1}{\left(\displaystyle \frac{{\rho }_{\mathrm{nuc}}0.5}{{m}_{{\rm{B}}}}\right)}^{{\gamma }_{1}-{\gamma }_{2}}{\left(\displaystyle \frac{{m}_{{\rm{B}}}}{{\rho }_{0}}\right)}^{{\gamma }_{1}-4/3}.\end{eqnarray} \tag{ 38 }$$

The entropy of the system is given by

$$\begin{eqnarray}&&S=-{\left(\displaystyle \frac{\partial F}{\partial T}\right)}_{V,{N}_{i}}={S}_{{\rm{n}}}+{S}_{{\rm{p}}},\end{eqnarray} \tag{ 39 }$$

where ${S}_{{\rm{i}}}$, ${\rm{i}}={\rm{n}},{\rm{p}}$ is given by

$$\begin{eqnarray}&&{S}_{{\rm{i}}}={N}_{{\rm{i}}}k\left[\mathrm{ln}\left\{\displaystyle \frac{{{Vg}}_{{\rm{i}}}}{{N}_{{\rm{n}}}}{\left(\displaystyle \frac{2\pi {m}_{{\rm{B}}}{kT}}{{h}^{2}}\right)}^{{E}_{\mathrm{coef}}}\right\}+{E}_{\mathrm{coef}}+1\right].\end{eqnarray} \tag{ 40 }$$

The internal energy of the system is given by

$$\begin{eqnarray}\begin{array}{rcl}E & = & F+{TS}={F}_{1}{\left(\displaystyle \frac{\rho {Y}_{{\rm{e}}}}{{m}_{{\rm{B}}}}\right)}^{{\gamma }_{1}}{\left(\displaystyle \frac{{m}_{{\rm{B}}}}{{\rho }_{0}}\right)}^{{\gamma }_{1}-4/3}V+{F}_{2}{\left(\displaystyle \frac{\rho {X}_{{\rm{n}}}}{{m}_{{\rm{B}}}}\right)}^{{\gamma }_{2}}V+{E}_{\mathrm{coef}}{N}_{{\rm{B}}}{kT}.\end{array}\end{eqnarray} \tag{ 41 }$$

Lastly, the chemical potentials are given by

$$\begin{eqnarray}\begin{array}{rcl}{\mu }_{{\rm{n}}} & = & -{\left(\displaystyle \frac{\partial F}{\partial {N}_{{\rm{n}}}}\right)}_{V,{N}_{{\rm{p}}},{N}_{{\rm{e}}}}={\gamma }_{2}{F}_{2}{\left(\displaystyle \frac{\rho {X}_{{\rm{n}}}}{{m}_{{\rm{B}}}}\right)}^{{\gamma }_{2}-1}\,+\,{kT}\mathrm{ln}\left[\displaystyle \frac{\rho {X}_{{\rm{n}}}}{{g}_{{\rm{n}}}{m}_{{\rm{B}}}}{\left(\displaystyle \frac{{h}^{2}}{2\pi {m}_{{\rm{B}}}{kT}}\right)}^{{E}_{\mathrm{coef}}}\right],\end{array}\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}&&{\mu }_{{\rm{p}}}=-{\left(\displaystyle \frac{\partial F}{\partial {N}_{{\rm{p}}}}\right)}_{V,{N}_{{\rm{n}}},{N}_{{\rm{e}}}}={kT}\mathrm{ln}\left[\displaystyle \frac{\rho {X}_{{\rm{p}}}}{{g}_{{\rm{p}}}{m}_{{\rm{B}}}}{\left(\displaystyle \frac{{h}^{2}}{2\pi {m}_{{\rm{B}}}{kT}}\right)}^{{E}_{\mathrm{coef}}}\right],\end{eqnarray} \tag{ 43 }$$

and

$$\begin{eqnarray}&&{\mu }_{{\rm{e}}}=-{\left(\displaystyle \frac{\partial F}{\partial {N}_{{\rm{e}}}}\right)}_{V,{N}_{{\rm{n}}},{N}_{{\rm{p}}}}={\gamma }_{1}{F}_{1}{\left(\displaystyle \frac{\rho {Y}_{{\rm{e}}}}{{m}_{{\rm{B}}}}\right)}^{{\gamma }_{1}-1}{\left(\displaystyle \frac{{m}_{{\rm{B}}}}{{\rho }_{0}}\right)}^{{\gamma }_{1}-4/3}.\end{eqnarray} \tag{ 44 }$$

That all of the thermodynamic functions are derived from an analytic thermodynamic potential, namely, the free energy, insures that the EoS is thermodynamically consistent. Furthermore, being completely analytic and simple, the interpolation scheme described above is not necessary when using this EoS. Derivatives of thermodynamic functions can be obtained analytically, and expressions for the independent variables, ρ, T, and ${Y}_{{\rm{e}}}$ can be solved for directly without having to invert a complicated expression by resorting to an iteration scheme.

The method of solution of the hydrodynamics equations is a finite-volume method, wherein conserved quantities are represented as integrated over computational volumes, or zones, and changes to these variables occur by means of sources or sinks or by means of fluxes through zone boundaries due to the relative velocity between the fluid and these boundaries. Chimera uses a Lagrangian-plus-remap version of the PPM method, which can be described by considering the zone-integrated conserved quantity, $q({\boldsymbol{x}},t)$, integrated over a computational element, ΔV(t), whose boundaries may be time-dependent

$$\begin{eqnarray}&&Q({\boldsymbol{x}},t)={\displaystyle \int }_{{\rm{\Delta }}V(t)}q({\boldsymbol{x}},t)\,{dV}.\end{eqnarray} \tag{ 45 }$$

The time rate of change of Q is

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{dQ}}{{dt}} & = & \displaystyle \frac{d}{{dt}}{\displaystyle \int }_{{\rm{\Delta }}V(t)}q({\boldsymbol{x}},t){dV}\\ & = & {\displaystyle \int }_{{\rm{\Delta }}V(t)}\displaystyle \frac{\partial q({\boldsymbol{x}},t)}{\partial t}\,{dV}+{\displaystyle \int }_{{\rm{\Delta }}S(t)}q({\boldsymbol{x}},t){{\boldsymbol{u}}}_{g}\cdot {\boldsymbol{n}}\,{dS}\\ & = & -{\displaystyle \int }_{{\rm{\Delta }}S(t)}q({\boldsymbol{x}},t){\boldsymbol{u}}\cdot {\boldsymbol{n}}\,{dS}+{\displaystyle \int }_{{\rm{\Delta }}V(t)}{{ \mathcal S }}_{q}({\boldsymbol{x}},t){dV}\,+\,{\displaystyle \int }_{{\rm{\Delta }}S(t)}q({\boldsymbol{x}},t){{\boldsymbol{u}}}_{g}\cdot {\boldsymbol{n}}\,{dS}\\ & = & -{\displaystyle \int }_{{\rm{\Delta }}S(t)}q({\boldsymbol{x}},t)\left({\boldsymbol{u}}-{{\boldsymbol{u}}}_{g}\right)\cdot {\boldsymbol{n}}\,{dS}+{\displaystyle \int }_{{\rm{\Delta }}V(t)}{{ \mathcal S }}_{q}({\boldsymbol{x}},t)\,{dV},\end{array}\end{eqnarray} \tag{ 46 }$$

where the second line results from the use of the Reynolds transport theorem, with ${{\boldsymbol{u}}}_{g}$, which we will refer to as the grid velocity, being the velocity of the bounding surface ${\rm{\Delta }}S(t)$ of the zone volume ${\rm{\Delta }}V(t)$. The first term in the third line is the flux of $q({\boldsymbol{x}},t)$ across the surface ΔS(t) due to the fluid velocity ${\boldsymbol{u}}$, assuming linear flux and the surface ΔS(t) to be fixed, and the second term represents a possible volume source ${{ \mathcal S }}_{q}({\boldsymbol{x}},t)$ of $q({\boldsymbol{x}},t)$. To calculate ${dQ}/{dt}$, Chimera splits the calculation into two steps,

$$\begin{eqnarray}&&\left(\displaystyle \frac{{dQ}}{{dt}}\right)={\left(\displaystyle \frac{{dQ}}{{dt}}\right)}^{\mathrm{step}1}+{\left(\displaystyle \frac{{dQ}}{{dt}}\right)}^{\mathrm{step}2}.\end{eqnarray} \tag{ 47 }$$

In "step 1," the grid velocity is set equal to the fluid velocity and the resulting equation

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{dQ}}{{dt}}\right)}^{\mathrm{step}1}={\displaystyle \int }_{{\rm{\Delta }}V(t)}{{ \mathcal S }}_{q}({\boldsymbol{x}},t){dV}\end{eqnarray} \tag{ 48 }$$

is solved. This is just the Lagrangian form of the equation for $\partial Q({\boldsymbol{x}},t)/\partial t$. In "step 2," the fluid velocity is set to zero and the initial configuration of ${{ \mathcal S }}_{q}({\boldsymbol{x}},t)$ is that of its final configuration after "step 1." The grid is then given a prescribed velocity so that

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{dQ}}{{dt}}\right)}^{\mathrm{step}2}={\displaystyle \int }_{{\rm{\Delta }}S(t)}q({\boldsymbol{x}},t){{\boldsymbol{u}}}_{g}\cdot {\boldsymbol{n}}\,{dS}.\end{eqnarray} \tag{ 49 }$$

If the grid velocity, ${{\boldsymbol{u}}}_{g}$, is chosen to be the negative of the fluid velocity, ${\boldsymbol{u}}$, then "step 1" plus "step 2" would be equivalent to

$$\begin{eqnarray}&&\displaystyle \frac{{dQ}}{{dt}}=-{\displaystyle \int }_{{\rm{\Delta }}S(t)}q({\boldsymbol{x}},t){\boldsymbol{u}}\cdot {\boldsymbol{n}}\,{dS}+{\displaystyle \int }_{{\rm{\Delta }}V(t)}{{ \mathcal S }}_{q}({\boldsymbol{x}},t){dV},\end{eqnarray} \tag{ 50 }$$

which is just the Eulerian form of the equation for $\partial Q({\boldsymbol{x}},t)/\partial t$. For the θ- and ϕ-grid, we choose to set ${{\boldsymbol{u}}}_{g}=-{\boldsymbol{u}}$ to keep these grids stationary. For the radial grid, however, we use the freedom of choice for ${{\boldsymbol{u}}}_{g}$ to make the grid dynamically adaptive, allowing it to move in such a way as to maintain good resolution during such epochs as core collapse or the formation of a density cliff in the vicinity of the neutrinospheres.

The solution of the hydrodynamics equations proceeds in a dimensionally split manner. We will describe the solution as it proceeds in a particular, but otherwise arbitrary, dimension and will refer to a specific dimension (e.g., r, θ, or ϕ) only when expressions specific to this dimension arise. In order to construct the finite difference representations of the underlying partial differential equations that Chimera solves, a discrete grid is set up dividing the interval ξ_min to ξ_max into a total of I zones, where ξ is the parameter (r, θ, or ϕ) specifying the coordinate distance along a ray in a given dimension. Figure 12 illustrates our indexing convection. At each end of the grid, six ghost zones are appended to hold boundary values of the quantities stored in the real zones 1... I. In a given dimension, both the Lagrangian and the remap steps in PPMLR hydrodynamics begin by constructing zone interface values of primitive quantities, such as ρ, p, and the components of ${\boldsymbol{u}}$, from zone-average values of these quantities. Lufkin & Hawley (1993) have shown that a differencing scheme that uses zone averages to represent fluid variables will not converge to the continuity equation (essential for conserving quantities during advection) unless the interpolation scheme, a(ξ), for the zone-averaged quantities, a_i, satisfies

$$\begin{eqnarray}&&{a}_{i}=\displaystyle \frac{1}{{V}_{i}}{\displaystyle \int }_{{\xi }_{i-\tfrac{1}{2}}}^{{\xi }_{i+\tfrac{1}{2}}}a(\xi )\displaystyle \frac{{dV}}{d\xi }d\xi ,\end{eqnarray} \tag{ 51 }$$

where V_i is the volume of zone i, and ${dV}/d\xi$ is the one-dimensional Jacobian determinant for V.

The interpolation scheme used here is the same as that described in Colella & Woodward (1984), as modified by Blondin & Lufkin (1993) to accommodate curvilinear coordinates while at the same time representing a linear velocity field accurately near coordinate boundaries—e.g., r = 0 in the radial dimension. The procedure for determining the zone interface value at i − $\tfrac{1}{2}$ of the zone-averaged variables, a_i, is to construct a quartic polynomial, $A(\xi )$, for each zone interface i − $\tfrac{1}{2}$ such that it takes on the respective values, ${A}_{i-\tfrac{5}{2}}\cdots {A}_{i+\tfrac{3}{2}}$ at the five points ${\xi }_{i-\tfrac{5}{2}}\cdots {\xi }_{i+\tfrac{3}{2}}$, where

$$\begin{eqnarray}&&{A}_{i-\tfrac{1}{2}}=\sum _{k\lt i}{a}_{k}{\rm{\Delta }}{V}_{k}.\end{eqnarray} \tag{ 52 }$$

The desired interpolating polynomial, a(ξ), is given by the integrand of the indefinite integral

$$\begin{eqnarray}&&A(\xi )={\displaystyle \int }^{\xi }a(\xi ^{\prime} ){dV}(\xi ^{\prime} )={\displaystyle \int }^{\xi }a(\xi ^{\prime} )\displaystyle \frac{{dV}}{d\xi ^{\prime} }d\xi ^{\prime} ;\end{eqnarray} \tag{ 53 }$$

that is,

$$\begin{eqnarray}&&a(\xi )=\displaystyle \frac{{dA}(\xi )}{d\xi }{\left(\displaystyle \frac{{dV}}{d\xi }\right)}^{-1}.\end{eqnarray} \tag{ 54 }$$

By construction, each cubic polynomial a(ξ) so obtained has the desired property

$$\begin{eqnarray}&&{a}_{j}{\rm{\Delta }}{V}_{j}={\displaystyle \int }_{{\xi }_{j-\tfrac{1}{2}}}^{{\xi }_{j+\tfrac{1}{2}}}a(\xi )\displaystyle \frac{{dV}}{d\xi ^{\prime} }d\xi ^{\prime} \quad j=i-2,\cdots ,\ i+1.\end{eqnarray} \tag{ 55 }$$

The interface value, ${a}_{i-\tfrac{1}{2}}$, is obtained from Equation (54) by evaluating it at $\xi ={\xi }_{i-\tfrac{1}{2}}$:

$$\begin{eqnarray}&&{a}_{i-\tfrac{1}{2}}={\left.\displaystyle \frac{{dA}(\xi )}{d\xi }\right|}_{i-\tfrac{1}{2}}{\left({\left.\displaystyle \frac{{dV}}{d\xi }\right|}_{i-\tfrac{1}{2}}\right)}^{-1},\end{eqnarray} \tag{ 56 }$$

where the explicit expression for A(ξ) is given by Equations (12) and (13) in Blondin & Lufkin (1993).

The interpolation for ${a}_{i-\tfrac{1}{2}}$ given by Blondin & Lufkin (1993) differs from Equations (1.6) and (1.7) of Colella & Woodward (1984). In the former publication, the zone-averaged quantities a_i are multiplied by the geometry-dependent correction factors ${\rm{\Delta }}{V}_{i}/{\rm{\Delta }}{\xi }_{i}$. Denoting these geometry corrected quantities by ${a}_{i}^{* }$, we have

$$\begin{eqnarray}&&{a}_{i}^{* }={a}_{i}\displaystyle \frac{{\rm{\Delta }}{V}_{i}}{{\rm{\Delta }}{\xi }_{i}},\end{eqnarray} \tag{ 57 }$$

where the geometry-dependent correction factor ${\rm{\Delta }}{V}_{i}/{\rm{\Delta }}{\xi }_{i}$ is given by

$$\begin{eqnarray}\displaystyle \frac{{\rm{\Delta }}{V}_{i}}{{\rm{\Delta }}{\xi }_{i}}=\left\{\begin{array}{cc}\displaystyle \frac{1}{3}\left({\xi }_{i-\tfrac{1}{2}}^{2}+{\xi }_{i-\tfrac{1}{2}}{\xi }_{i+\tfrac{1}{2}}+{\xi }_{i+\tfrac{1}{2}}^{2}\right), & r{\rm{ \mbox{-} direction}}\\ \left(\cos {\xi }_{i-\tfrac{1}{2}}-\cos {\xi }_{i+\tfrac{1}{2}}\right)/{\rm{\Delta }}{\xi }_{i}, & \theta {\rm{ \mbox{-} direction}}\\ 1, & \phi {\rm{ \mbox{-} direction}}\end{array}\right.\end{eqnarray} \tag{ 58 }$$

and where

$$\begin{eqnarray}&&{\rm{\Delta }}{\xi }_{i}={\xi }_{i+\tfrac{1}{2}}-{\xi }_{i-\tfrac{1}{2}}.\end{eqnarray} \tag{ 59 }$$

The average slope, $\delta {a}_{i}^{* }$, of the parabolas are then computed from Equation (1.7) of Colella & Woodward (1984) but using the quantities ${a}_{i}^{* };$ e.g.,

$$\begin{eqnarray}\begin{array}{rcl}\delta {a}_{i}^{* } & = & \displaystyle \frac{{\rm{\Delta }}{\xi }_{i}}{{\rm{\Delta }}{\xi }_{i-1}+{\rm{\Delta }}{\xi }_{i}+{\rm{\Delta }}{\xi }_{i+1}}\,\left[\displaystyle \frac{2{\rm{\Delta }}{\xi }_{i-1}+{\rm{\Delta }}{\xi }_{i}}{{\rm{\Delta }}{\xi }_{i+1}+{\rm{\Delta }}{\xi }_{i}}{\rm{\Delta }}{a}_{i+\tfrac{1}{2}}^{* }+\displaystyle \frac{{\rm{\Delta }}{\xi }_{i}+2{\rm{\Delta }}{\xi }_{i+1}}{{\rm{\Delta }}{\xi }_{i-1}+{\rm{\Delta }}{\xi }_{i}}{\rm{\Delta }}{a}_{i-\tfrac{1}{2}}^{* }\right],\end{array}\end{eqnarray} \tag{ 60 }$$

where

$$\begin{eqnarray}&&{\rm{\Delta }}{a}_{i-\tfrac{1}{2}}^{* }={a}_{i}^{* }-{a}_{i-1}^{* }.\end{eqnarray} \tag{ 61 }$$

The $\delta {a}_{i}^{* }$ are then modified as follows (see; Colella & Woodward 1984, Equation (1.8)):

$$\begin{eqnarray}{\delta }_{m}{a}_{i}^{* }=\left\{\begin{array}{cc}\min \left(\left|\delta {a}_{i}^{* }\right|,2\left|{a}_{i+1}^{* }-{a}_{i}^{* }\right|,2\left|{a}_{i}^{* }-{a}_{i-1}^{* }\right|\right){\rm{}}\,{\rm{sign}}(\delta {a}_{i}^{* }), & {\rm{}}\,{\rm{if}}\ {\rm{\Delta }}{a}_{i+\tfrac{1}{2}}^{* }{\rm{\Delta }}{a}_{i-\tfrac{1}{2}}^{* }\gt 0,\\ 0 & {\rm{}}\,{\rm{if}}\ {\rm{\Delta }}{a}_{i+\tfrac{1}{2}}^{* }{\rm{\Delta }}{a}_{i-\tfrac{1}{2}}^{* }\lt 0.\end{array}\right.\end{eqnarray} \tag{ 62 }$$

The modifications for both cases implement monotonicity constraints, ensuring that no new maxima or minima appear (i.e., that ${a}_{i-\tfrac{1}{2}}$ lies in the range of ${a}_{i}^{* }$ and ${a}_{i-1}^{* }$), and in the case of ${\rm{\Delta }}{a}_{i+\tfrac{1}{2}}^{* }{\rm{\Delta }}{a}_{i-\tfrac{1}{2}}^{* }\gt 0$ lead to somewhat steeper representations of discontinuities.

Given ${a}_{i}^{* }$ and ${\delta }_{m}{a}_{i}^{* }$, the interface value ${a}_{i+\tfrac{1}{2}}$ is now obtained from the cubic interpolating polynomial, Equation (54), which is Equation (1.6) of Colella & Woodward (1984) with the geometry-dependent corrections applied as above in Equation (57) and below in Equation (64):

$$\begin{eqnarray}\begin{array}{rcl}{a}_{i+\tfrac{1}{2}}^{* } & = & {a}_{i}^{* }+\displaystyle \frac{{\rm{\Delta }}{\xi }_{i}}{{\rm{\Delta }}{\xi }_{i}+{\rm{\Delta }}{\xi }_{i+1}}\left({a}_{i+1}^{* }-{a}_{i}^{* }\right)\\ & & +\,\displaystyle \frac{1}{{\sum }_{k=-1}^{2}{\rm{\Delta }}{\xi }_{i+k}}\left\{\displaystyle \frac{2{\rm{\Delta }}{\xi }_{i+1}{\rm{\Delta }}{\xi }_{i}}{{\rm{\Delta }}{\xi }_{i}{\rm{\Delta }}{\xi }_{i+1}}\left[\displaystyle \frac{{\rm{\Delta }}{\xi }_{i-1}+{\rm{\Delta }}{\xi }_{i}}{2{\rm{\Delta }}{\xi }_{i}+{\rm{\Delta }}{\xi }_{i+1}}\right.\right.\left.\,-\displaystyle \frac{{\rm{\Delta }}{\xi }_{i+1}+{\rm{\Delta }}{\xi }_{i+2}}{{\rm{\Delta }}{\xi }_{i}+2{\rm{\Delta }}{\xi }_{i+1}}\right]\left({a}_{i+1}^{* }-{a}_{i}^{* }\right)\\ & & -{\rm{\Delta }}{\xi }_{i}\displaystyle \frac{{\rm{\Delta }}{\xi }_{i-1}+{\rm{\Delta }}{\xi }_{i}}{2{\rm{\Delta }}{\xi }_{i}+{\rm{\Delta }}{\xi }_{i+1}}\delta {a}_{m,i+1}^{* }+\,\left.{\rm{\Delta }}{\xi }_{i+1}\displaystyle \frac{{\rm{\Delta }}{\xi }_{i+1}+{\rm{\Delta }}{\xi }_{i+2}}{{\rm{\Delta }}{\xi }_{i}+2{\rm{\Delta }}{\xi }_{i+1}}\delta {a}_{m,i}^{* }\right\},\end{array}\end{eqnarray} \tag{ 63 }$$

$$\begin{eqnarray}&&{a}_{i+\tfrac{1}{2}}={a}_{i+\tfrac{1}{2}}^{* }{\left(\displaystyle \frac{\partial V}{\partial \xi }\right)}_{i+\tfrac{1}{2}}^{-1},\end{eqnarray} \tag{ 64 }$$

where

$$\begin{eqnarray}{\left(\displaystyle \frac{\partial V}{\partial \xi }\right)}_{i+\tfrac{1}{2}}=\left\{\begin{array}{cc}{\xi }_{i+\tfrac{1}{2}}^{2}\quad (1{\rm{}}\,{\rm{if}}{\xi }_{\tfrac{1}{2}}=0), & r{\rm{ \mbox{-} direction}}\\ \sin {\xi }_{i+\tfrac{1}{2}}\quad (1{\rm{}}\,{\rm{if}}{\xi }_{\tfrac{1}{2}}=0), & \theta {\rm{ \mbox{-} direction}}\\ 1, & \phi {\rm{ \mbox{-} direction}}\end{array}\right.\end{eqnarray} \tag{ 65 }$$

where the values at ${\xi }_{\tfrac{1}{2}}=0$ are to avoid singularities in Equation (64). Finally, the range of ${a}_{R,i}$ is limited to be within the range of a_i and ${a}_{i+1}$:

$$\begin{eqnarray}&&{a}_{R,i}=\max ({a}_{i+\tfrac{1}{2}},\min ({a}_{i},{a}_{i+1})),\end{eqnarray} \tag{ 66 }$$

$$\begin{eqnarray}&&{a}_{R,i}=\min ({a}_{R,i},\max ({a}_{i},{a}_{i+1})),\end{eqnarray} \tag{ 67 }$$

and

$$\begin{eqnarray}&&{a}_{L,i+1}={a}_{R,i},\end{eqnarray} \tag{ 68 }$$

where ${a}_{L,i+1}$ is the value of a(ξ) at the left interface of zone i + 1, and ${a}_{R,i}$ is its value at the right interface of zone i. At zone boundaries, ${a}_{L,i}$ and ${a}_{R,i}$ must be modified in certain cases as follows. In the r-direction, if ${\xi }_{\tfrac{1}{2}}=0$, then ${a}_{L,i}$ must be modified in a manner depending on whether the variable is odd (e.g., velocity) or even (e.g., density, specific energy) at the origin:

$$\begin{eqnarray}\begin{array}{lll}{a}_{R,0}=0, & \,{a}_{L,1}=0, & \,{\rm{odd}}\,{\rm{variables}}\,{\rm{at}}\,r=0,\\ {a}_{R,0}=\displaystyle \frac{5}{2}{a}_{1}-\displaystyle \frac{3}{2}{a}_{L,2}, & \,{a}_{L,1}={a}_{R,0} & \,{\rm{even}}\,{\rm{variables}}\,{\rm{at}}\,r=0.\end{array}\end{eqnarray} \tag{ 69 }$$

In the θ-direction, if ${\xi }_{\tfrac{1}{2}}=0(\theta =0)$ and reflecting boundary conditions are imposed,

$$\begin{eqnarray}\begin{array}{lll}{a}_{R,0}=0, & \,{a}_{L,1}=0, & \,{\rm{odd}}\,{\rm{variables}}\,{\rm{at}}\ \theta =0,\\ {a}_{R,0}=\left(6\,{a}_{1}+{a}_{L,2}\right)/7, & \,{a}_{L,1}={a}_{R,0} & \,{\rm{even}}\,{\rm{variables}}\,{\rm{at}}\ \theta =0,\end{array}\end{eqnarray} \tag{ 70 }$$

and if ${\xi }_{I+\tfrac{1}{2}}=0(\theta =\pi )$,

$$\begin{eqnarray}\begin{array}{lll}{a}_{R,I} & = & 0,\,{a}_{L,I+1}=0,\,{\rm{odd}}\,{\rm{variables}}\,{\rm{at}}\ \theta =\pi ,\\ {a}_{R,I} & = & \left(6\,{a}_{I}+{a}_{L,I-1}\right)/7,\,{a}_{L,I+1}={a}_{R,I}\,{\rm{even}}\,{\rm{variables}}\,{\rm{at}}\,\theta =\pi .\end{array}\end{eqnarray} \tag{ 71 }$$

In the presence of shocks, post-shock oscillations sometimes occur in some of the fluid variables, e.g., entropy (Colella & Woodward 1984, Section 4). One method of suppressing these oscillations is to introduce some additional dissipation in the vicinity of a shock. The method used here is to lower the order of the interpolation (i.e., flatten the interpolation profile) in the vicinity of a shock. Thus, in the vicinity of a shock, a_L,I and a_R,I are modified as follows:

$$\begin{eqnarray}\begin{array}{rcl}{a}_{L,I} & = & {a}_{i}f+{a}_{L,I}(1-f),\\ {a}_{R,I} & = & {a}_{i}f+{a}_{R,I}(1-f),\end{array}\end{eqnarray} \tag{ 72 }$$

where the "flattening parameter," 0 ≤ f ≤ 1, is zero away from shocks and approaches a user preset value f_set ≤ 1 in the limit of a strong shock with a steep profile.

The flattening parameter f_i is computed similarly to that described in Fryxell et al. (2000) and begins with the computation of the pressure profile ${p}_{\mathrm{profile}i}$ given by

$$\begin{eqnarray}&&{p}_{\mathrm{profile}i}=\displaystyle \frac{{p}_{i+1}-{p}_{i-1}}{{p}_{i+2}-{p}_{i-2}}.\end{eqnarray} \tag{ 73 }$$

The presence of a shock is indicated if this quantity approaches 1, i.e., when ${p}_{i+2}-{p}_{i-2}\approx {p}_{i+1}-{p}_{i-1}$. For smooth flows, ${p}_{i+2}-{p}_{i-2}\approx 2\times ({p}_{i+1}-{p}_{i-1})$ and ${p}_{\mathrm{profile}i}\approx 1/2$. A pre-flattening parameter ${f}_{\mathrm{pre}i}$ is then computed from

$$\begin{eqnarray}&&{f}_{\mathrm{pre}i}=\max \left[\left({p}_{\mathrm{profile}i}-{\omega }_{1}\right)\times {\omega }_{2},0\right],\end{eqnarray} \tag{ 74 }$$

where ω₁ and ω₁ are user selected parameters. We have found that 0.75 and 10 work well for ω₁ and ω₂ during the early stages of a simulation when the shock is nearly spherically symmetric, completely eliminating post-shock oscillations, and 0.6 and 10 work well thereafter better capturing oblique shocks. The pre-flattening parameter is set to zero if

$$\begin{eqnarray}&&\displaystyle \frac{\left|{p}_{i+1}-p+i-1\right|}{\min \left[\max \left({p}_{i+1},0.05{p}_{i-1}\right),\max \left({p}_{i-1},0.05{p}_{i+1}\right)\right]}\lt \displaystyle \frac{1}{3},\end{eqnarray} \tag{ 75 }$$

which indicates an insufficient pressure jump, and if

$$\begin{eqnarray}&&{u}_{i-1}\lt {u}_{i+1},\end{eqnarray} \tag{ 76 }$$

which indicates a velocity jump in the wrong direction. Finally, the flattening parameter is given by

$$\begin{eqnarray}&&{f}_{i}=\max \left\{0,\min \left[{f}_{\mathrm{set}},\max \left({f}_{\mathrm{pre}i-1},{f}_{\mathrm{pre}i},{f}_{\mathrm{pre}i+1}\right)\right]\right\}.\end{eqnarray} \tag{ 77 }$$

This limits the value of f_i to $0\leqslant {f}_{\mathrm{set}}$ and sets the value f_i based on the maximum value of ${f}_{\mathrm{pre}}$ in zone i and the neighboring zones. Experimentation has indicated that f_set = 1 works well when the shock is approximately spherically symmetric, and f_set = 0.5 works well thereafter.

With the values of ${a}_{L,i}$ and ${a}_{R,i}$ for each zone i determined, a piecewise parabolic interpolation function, a(ξ), is constructed with a(ξ) given by a parabolic profile in each zone:

$$\begin{eqnarray}\begin{array}{rclrcl}a(\xi ) & = & {a}_{L,i}+x({\rm{\Delta }}{a}_{i}+{a}_{6,i}(1-x)) & x & = & \displaystyle \frac{\xi -{\xi }_{i-\tfrac{1}{2}}}{{\xi }_{i+\tfrac{1}{2}}-{\xi }_{i-\tfrac{1}{2}}},\quad {\xi }_{i-\tfrac{1}{2}}\leqslant \xi \leqslant {\xi }_{i+\tfrac{1}{2}},\end{array}\end{eqnarray} \tag{ 78 }$$

where

$$\begin{eqnarray}&&{\rm{\Delta }}{a}_{i}={a}_{R,i}-{a}_{L,i},\end{eqnarray} \tag{ 79 }$$

and where ${a}_{6,i}$. The parameter ${a}_{6,i}$ must now be determined so that Equation (51) is satisfied. The expressions for ${a}_{6,i}$ are given by

$$\begin{eqnarray}{a}_{6,i}=\left\{\begin{array}{cc}6\left\{{a}_{i}-\displaystyle \frac{1}{2}\left[{a}_{L,i}\left(1-{{ \mathcal F }}_{r}\right)+{a}_{R,i}\left(1+{{ \mathcal F }}_{r}\right)\right]\right\}{{ \mathcal G }}_{r} & r{\rm{ \mbox{-} direction}},\\ \left[{a}_{i}\left({{ \mathcal G }}_{\theta }-{{ \mathcal F }}_{\theta }\right)-{a}_{L,i}{{ \mathcal G }}_{\theta }+{a}_{R,i}{{ \mathcal F }}_{\theta }\right]/{{ \mathcal H }}_{\theta } & \theta {\rm{ \mbox{-} direction}},\\ 6\left[{a}_{i}-\displaystyle \frac{1}{2}\left({a}_{L,i}+{a}_{R,i}\right)\right] & \phi {\rm{ \mbox{-} direction}},\end{array}\right.\end{eqnarray} \tag{ 80 }$$

where

$$\begin{eqnarray}&&{{ \mathcal F }}_{r}=\left(y+1/2\right)/\left(3{y}^{2}+3y+1\right),\end{eqnarray} \tag{ 81 }$$

$$\begin{eqnarray}&&{{ \mathcal G }}_{r}=\left(3{y}^{2}+3y+1\right)/\left(3{y}^{2}+3y+9/10\right),\end{eqnarray} \tag{ 82 }$$

$$\begin{eqnarray}&&y={\xi }_{i-\tfrac{1}{2}}/{\rm{\Delta }}{\xi }_{i},\end{eqnarray} \tag{ 83 }$$

and

$$\begin{eqnarray}&&{{ \mathcal F }}_{\theta }=\cos \left({\xi }_{i+\tfrac{1}{2}}\right)-\left[\sin \left({\xi }_{i+\tfrac{1}{2}}\right)-\sin \left({\xi }_{i-\tfrac{1}{2}}\right)\right]/{\rm{\Delta }}{x}_{i},\end{eqnarray} \tag{ 84 }$$

$$\begin{eqnarray}&&{{ \mathcal G }}_{\theta }=\cos \left({\xi }_{i-\tfrac{1}{2}}\right)-\left[\sin \left({\xi }_{i+\tfrac{1}{2}}\right)-\sin \left({\xi }_{i-\tfrac{1}{2}}\right)\right]/{\rm{\Delta }}{x}_{i},\end{eqnarray} \tag{ 85 }$$

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal H }}_{\theta } & = & 2\left[\cos \left({\xi }_{i-\tfrac{1}{2}}\right)-\cos \left({\xi }_{i+\tfrac{1}{2}}\right)\right]/{\left({\rm{\Delta }}{x}_{i}\right)}^{2}-\left[\sin \left({\xi }_{i+\tfrac{1}{2}}\right)-\sin \left({\xi }_{i-\tfrac{1}{2}}\right)\right]/{\rm{\Delta }}{x}_{i}.\end{array}\end{eqnarray} \tag{ 86 }$$

In addition to the constraints imposed by Equations (62) and (66)–(68), the following monotonicity constraints are imposed to avoid the possibility of the interpolating function taking on values not between ${a}_{L,i}$ and ${a}_{R,i}$, which could otherwise lead to its developing spurious oscillations:

$$\begin{eqnarray}{a}_{L,i,\mathrm{mon}}=\left\{\begin{array}{cc}{a}_{R,i}\left(1+{{ \mathcal H }}_{r}^{+}\right)-{a}_{i}{{ \mathcal H }}_{r}^{+} & r{\rm{ \mbox{-} direction}},\\ \left[{a}_{R,i}\left({{ \mathcal F }}_{\theta }-{{ \mathcal H }}_{\theta }\right)+{a}_{i}\left({{ \mathcal G }}_{\theta }-{{ \mathcal F }}_{\theta }\right)\right]/\left({{ \mathcal G }}_{\theta }-{{ \mathcal H }}_{\theta }\right) & \theta {\rm{ \mbox{-} direction}},\\ 3{a}_{i}-2{a}_{R,i} & \phi {\rm{ \mbox{-} direction}},\end{array}\right.\end{eqnarray} \tag{ 87 }$$

$$\begin{eqnarray}{a}_{R,i,\mathrm{mon}}=\left\{\begin{array}{cc}{a}_{L,i}\left(1-{{ \mathcal H }}_{r}^{-}\right)+{a}_{i}{{ \mathcal H }}_{r}^{-} & r{\rm{ \mbox{-} direction}},\\ \left[{a}_{L,i}\left({{ \mathcal G }}_{\theta }-{{ \mathcal H }}_{\theta }\right)+{a}_{i}\left({{ \mathcal G }}_{\theta }-{{ \mathcal F }}_{\theta }\right)\right]/\left({{ \mathcal F }}_{\theta }-{{ \mathcal H }}_{\theta }\right) & \theta {\rm{ \mbox{-} direction}},\\ 3{a}_{i}-2{a}_{L,i} & \phi {\rm{ \mbox{-} direction}},\end{array}\right.\end{eqnarray} \tag{ 88 }$$

where

$$\begin{eqnarray}&&{{ \mathcal H }}_{r}^{+}=\displaystyle \frac{6}{3{{ \mathcal F }}_{r}+1/{{ \mathcal G }}_{r}-3},\qquad {{ \mathcal H }}_{r}^{-}=\displaystyle \frac{6}{3{{ \mathcal F }}_{r}-1/{{ \mathcal G }}_{r}+3}.\end{eqnarray} \tag{ 89 }$$

The need to use the monotonized expression for the left and right states arises if the parabola exceeds either of these states. With Δa_i given by Equation (79), we thus have

$$\begin{eqnarray}&&{a}_{L,i}={a}_{L,i,\mathrm{mon}}\quad {\rm{if}}\quad {\left({\rm{\Delta }}{a}_{i}\right)}^{2}\lt {\rm{\Delta }}{a}_{i}\,{a}_{6,i},\end{eqnarray} \tag{ 90 }$$

$$\begin{eqnarray}&&{a}_{R,i}={a}_{R,i,\mathrm{mon}}\quad {\rm{if}}\quad {\left({\rm{\Delta }}{a}_{i}\right)}^{2}\lt -{\rm{\Delta }}{a}_{i}\,{a}_{6,i}.\end{eqnarray} \tag{ 91 }$$

With these now monotonized values of ${a}_{L,i}$ and ${a}_{R,i}$, the quantities ${\rm{\Delta }}{a}_{i}$, and ${a}_{6,i}$ are recomputed from Equations (79) and (80). This completes the piecewise parabolic interpolation for the profile, a(ξ), of zone-averaged variables a_i.

The equations describing the change in the hydrodynamic variables during the Lagrangian step are:

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}{\displaystyle \int }_{V(t)}\rho \,{dV}=0,\end{eqnarray} \tag{ 92 }$$

$$\begin{eqnarray}\begin{array}{ll}\displaystyle \frac{d}{{dt}}{\displaystyle \int }_{V(t)}(\rho {\boldsymbol{u}}){dV}=\,-{\displaystyle \int }_{S(t)}p\hat{{\boldsymbol{n}}}\,{dS}+{\displaystyle \int }_{V(t)}\rho \left[-{\rm{\nabla }}{e}_{\mathrm{grav}}\right. & \left.\,+\,{{\boldsymbol{f}}}_{\nu }+{{\boldsymbol{f}}}_{\mathrm{ccor}}\right]{dV},\end{array}\end{eqnarray} \tag{ 93 }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d}{{dt}}{\displaystyle \int }_{V(t)}\rho ({e}_{\mathrm{int}}+{e}_{\mathrm{kin}}){dV} & = & -{\displaystyle \int }_{S(t)}p{\boldsymbol{u}}\cdot \hat{{\boldsymbol{n}}}\,{dS}\,+{\displaystyle \int }_{V(t)}\rho {\boldsymbol{u}}\cdot \left[-{\rm{\nabla }}{e}_{\mathrm{grav}}+{{\boldsymbol{f}}}_{\nu }\right]{dV},\end{array}\end{eqnarray} \tag{ 94 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}{\displaystyle \int }_{V(t)}(\rho {Y}_{{\rm{e}}}){dV}=\left[{\left.\displaystyle \frac{\partial {Y}_{{\rm{e}}}}{\partial t}\right|}_{\nu \mbox{-} \mathrm{interactions}}\right],\,{\rm{and}}\end{eqnarray} \tag{ 95 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}{\displaystyle \int }_{V(t)}(\rho {X}_{n}){dV}=\left[{\left.\displaystyle \frac{\partial {X}_{n}}{\partial t}\right|}_{\nu \mbox{-} \mathrm{interactions},\mathrm{nuclear}\mathrm{reactions}}\right],\end{eqnarray} \tag{ 96 }$$

where V(t) and S(t) are the volume and surface of a grid element or mass zone as it comoves with the fluid, $\hat{{\boldsymbol{n}}}$ is a unit vector normal to dS and pointing out of the mass zone, ${e}_{\mathrm{grav}}$ is the gravitational potential, ${{\boldsymbol{f}}}_{\nu }$ is the specific neutrino stress, and ${{\boldsymbol{f}}}_{\mathrm{ccor}}$ denotes the centrifugal and Coriolis forces. The time derivatives are taken at constant mass (i.e., are Lagrangian time derivatives). The expressions in the brackets on the right-hand sides of Equations (95) and (96) denote the changes in the electron fraction, ${Y}_{{\rm{e}}}$, due to neutrino transport and in the composition mass fractions, X_n, due to both neutrino transport and nuclear reactions, and are calculated elsewhere in the computational sweep, as described in Sections 6.8 and 3.3. During the Lagrangian hydrodynamics step, these expressions are set to zero.

Equation (94) is a common formulation of energy conservation on which difference schemes are subsequently constructed (e.g., Stone & Norman 1992; Bryan et al. 1995; Fryxell et al. 2000; Sutherland 2010). Chimera uses two alternative formulations of energy conservation, depending on the circumstances. Using Equations (92) and (93) in Equation (94), the following expression for ${e}_{\mathrm{int}}$ can be derived:

$$\begin{eqnarray}&&\displaystyle \frac{{{de}}_{\mathrm{int}}}{{dt}}=\displaystyle \frac{p}{\rho }{\left(\displaystyle \frac{\partial \rho }{\partial t}\right)}_{S,{Y}_{{\rm{e}}}},\end{eqnarray} \tag{ 97 }$$

which is just the first law in the absence of changes in entropy and electron fraction. This equation is applicable for updating the specific internal energy during the Lagrangian step in regions well away from shocks. It is more accurate numerically, in some cases, than updating the specific total energy as it does not ultimately involve the subtraction of potentially large values of the specific kinetic energy and the gravitational potential (if the latter is also included as a component in the specific total energy). In the vicinity of shocks, however, the specific total energy must be updated from the solution of the Riemann shock tube problem for the pressure and velocity at the zone interfaces. To do this, the term involving $\rho {\boldsymbol{u}}\cdot {\rm{\nabla }}{e}_{\mathrm{grav}}$ on the right-hand side of the energy Equation (94) is transformed as follows:

$$\begin{eqnarray}&&\begin{array}{l}{\displaystyle \int }_{V(t)}\rho {\boldsymbol{u}}\cdot \left[-{\rm{\nabla }}{e}_{\mathrm{grav}}\right]{dV}\,=\,{\displaystyle \int }_{V(t)}\left[-{\rm{\nabla }}\cdot \left(\rho {\boldsymbol{u}}{e}_{\mathrm{grav}}\right)+{e}_{\mathrm{grav}}{\rm{\nabla }}\cdot \left(\rho {\boldsymbol{u}}\right)\right]{dV}\\ \,=\,-{\displaystyle \int }_{V(t)}\left[{\rm{\nabla }}\cdot \left(\rho {\boldsymbol{u}}{e}_{\mathrm{grav}}\right)+{e}_{\mathrm{grav}}\displaystyle \frac{\partial \rho }{\partial t}\right]{dV}\\ \,=\,-{\displaystyle \int }_{V(t)}\left[{\rm{\nabla }}\cdot \left(\rho {\boldsymbol{u}}{e}_{\mathrm{grav}}\right)+\displaystyle \frac{\partial ({e}_{\mathrm{grav}}\rho )}{\partial t}-\rho \displaystyle \frac{\partial {e}_{\mathrm{grav}}}{\partial t}\right]{dV}\\ \,=\,-{\displaystyle \int }_{V(t)}\left[\displaystyle \frac{\partial ({e}_{\mathrm{grav}}\rho )}{\partial t}+{\boldsymbol{u}}\cdot {\rm{\nabla }}\left(\rho {e}_{\mathrm{grav}}\right)\right.+\left.\,\rho {e}_{\mathrm{grav}}{\rm{\nabla }}\cdot {\boldsymbol{u}}-\rho \displaystyle \frac{\partial {e}_{\mathrm{grav}}}{\partial t}\right]{dV}\\ \,=\,-\displaystyle \frac{d}{{dt}}{\displaystyle \int }_{V(t)}\rho {e}_{\mathrm{grav}}{dV}+{\displaystyle \int }_{V(t)}\rho \displaystyle \frac{\partial {e}_{\mathrm{grav}}}{\partial t}{dV}.\end{array}\end{eqnarray} \tag{ 98 }$$

Substituting Equation (98) in Equation (94) results in the following equation for the total energy (including gravitational energy)

$$\begin{eqnarray}\begin{array}{ll}\displaystyle \frac{d}{{dt}}{\displaystyle \int }_{V(t)}\rho ({e}_{\mathrm{int}}+{e}_{\mathrm{kin}}+{e}_{\mathrm{grav}}){dV} & =\,-{\displaystyle \int }_{S(t)}p{\boldsymbol{u}}\cdot \hat{{\boldsymbol{n}}}\,{dS}+{\displaystyle \int }_{V(t)}\left[\rho {\boldsymbol{u}}\cdot {{\boldsymbol{f}}}_{\nu }+\rho \displaystyle \frac{\partial {e}_{\mathrm{grav}}}{\partial t}\right]{dV}.\end{array}\end{eqnarray} \tag{ 99 }$$

Equation (99) is used in the radial-sweep to update the specific total energy in the vicinity of a shock. For the θ- and ϕ-sweeps, changes in the gravitational potential are very small, and Equation (94) is used with the gradient of the gravitational potential on the right-hand side treated as a force.

To perform the Lagrangian update, the first step is to compute the displacement of the zone interfaces, after which Equations (92), (93), (95), (96), and (97) or (99) are used to update ρ, the components of ${\boldsymbol{u}}$, ${Y}_{{\rm{e}}}$, and ${e}_{\mathrm{int}}$. The displacement of each zone interface during the Lagrangian step is determined by solving a Riemann problem for the velocity of the contact discontinuity at the zone interface. This requires averages of the needed quantities over the domains of dependences of the left and right states. Rather than solving the exact Riemann problem, which is time consuming, as it is complicated and involves multiple calls to the EoS, Chimera uses the approximate but very accurate method developed by Colella & Glaz (1985). This method parameterizes the EoS by the slowly varying quantity γ, given by

$$\begin{eqnarray}&&\gamma =\displaystyle \frac{p}{\rho {e}_{\mathrm{int}}}+1,\end{eqnarray} \tag{ 100 }$$

and the adiabatic exponent, Γ, defined by

$$\begin{eqnarray}&&{\rm{\Gamma }}={\left(\displaystyle \frac{\partial p}{\partial \rho }\right)}_{S,{Y}_{{\rm{e}}}}.\end{eqnarray} \tag{ 101 }$$

A solution for the approximate Riemann problem requires the values of the quantities ρ, u (the component of velocity in the direction of the directional splitting), p, γ, and Γ to the left and right of each zone interface. To maintain high-order accuracy, the values of each of these quantities are averaged over their domain of dependence of the zone interface as determined by the time step. This is accomplished for each zone interface by tracing the two characteristics from the interface at time t + Δt to the ξ axis at time t. Having speeds of ±c_s, where c_s is the local sound speed, the two characteristics intersect the ξ axis on either side of the interface at the points ${\xi }_{i+\tfrac{1}{2}}+{c}_{{\rm{s}}}{\rm{\Delta }}t$ and ${\xi }_{i+\tfrac{1}{2}}-{c}_{{\rm{s}}}{\rm{\Delta }}t$. Letting a_i represent any one of the above quantities, the average, $\langle a{\rangle }_{L,i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$, of a_i over the domain of dependence to the left of the zone interface, ${\xi }_{i+\tfrac{1}{2}}$, is obtained by integrating the parabolic profile $a(\xi )$ of a_i over the interval ${\xi }_{i+\tfrac{1}{2}}-{c}_{{\rm{s}},i}{\rm{\Delta }}t$ to ${\xi }_{i+\tfrac{1}{2}}$ and averaging, and is given by

$$\begin{eqnarray}&&\langle a{\rangle }_{L,i+\tfrac{1}{2}}={a}_{L,i}+{\rm{\Delta }}{a}_{i}-\displaystyle \frac{{c}_{{\rm{s}},i}{\rm{\Delta }}t}{2{\rm{\Delta }}{\xi }_{i}}\left[{\rm{\Delta }}{a}_{i}-\left(1-\displaystyle \frac{4}{3}\displaystyle \frac{{c}_{{\rm{s}},i}}{2{\rm{\Delta }}{\xi }_{i}}\right){a}_{6i}\right],\end{eqnarray} \tag{ 102 }$$

where Δa_i is given by Equation (79), Δξ_i is given by Equation (59), and ${c}_{{\rm{s}},i}^{2}={{\rm{\Gamma }}}_{i}{p}_{i}/{\rho }_{i}$. Likewise, the average, $\langle a{\rangle }_{R,i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$, of a over the domain of dependence to the right of the zone interface, ${\xi }_{i+\tfrac{1}{2}}$, is obtained by integrating the parabolic profile a(ξ) of a_i over the interval ${\xi }_{i+\tfrac{1}{2}}$ to ${\xi }_{i+\tfrac{1}{2}}+{c}_{{\rm{s}},i+1}{\rm{\Delta }}t$ and averaging, and is given by

$$\begin{eqnarray}\begin{array}{ll}\langle a{\rangle }_{R,i+\tfrac{1}{2}} & \ =\,{a}_{L,i+1}+\displaystyle \frac{{c}_{{\rm{s}},i+1}{\rm{\Delta }}t}{2{\rm{\Delta }}{\xi }_{i+1}}\left[{\rm{\Delta }}{a}_{i+1}+\left(1+\displaystyle \frac{4}{3}\displaystyle \frac{{c}_{{\rm{s}},i+1}{\rm{\Delta }}t}{2{\rm{\Delta }}{\xi }_{i+1}}\right){a}_{6,i+1}\right].\end{array}\end{eqnarray} \tag{ 103 }$$

The time-averaged left and right states, $\langle {p}^{+}{\rangle }_{L,i+\tfrac{1}{2}}$, $\langle {p}^{+}{\rangle }_{R,i+\tfrac{1}{2}}$, $\langle u{\rangle }_{L,i+\tfrac{1}{2}}$, $\langle u{\rangle }_{R,i+\tfrac{1}{2}}$ of p and u are obtained from Equations (102) and (103) above, and the time-averaged pressure is then corrected for the presence of gravitational, neutrino, centrifugal, and Coriolis forces by

$$\begin{eqnarray}\begin{array}{rcl}\langle p{\rangle }_{L,i+\tfrac{1}{2}} & = & \langle {p}^{+}{\rangle }_{L,i+\tfrac{1}{2}}+\displaystyle \frac{{\rm{\Delta }}t\,{\rho }_{i}\,{c}_{{\rm{s}},i}}{2}\left(-{\rm{\nabla }}{e}_{\mathrm{grav},i+\tfrac{1}{2}}\right.{\left.+{{\boldsymbol{f}}}_{\nu ,i+\tfrac{1}{2}}+{{\boldsymbol{f}}}_{\mathrm{ccor},{\rm{i}}+\tfrac{1}{2}}\right)}_{r,\theta ,\phi },\end{array}\end{eqnarray} \tag{ 104 }$$

$$\begin{eqnarray}\begin{array}{rcl}\langle p{\rangle }_{R,i+\tfrac{1}{2}} & = & \langle {p}^{+}{\rangle }_{R,i+\tfrac{1}{2}}-\displaystyle \frac{{\rm{\Delta }}t\,{\rho }_{i+1}\,{c}_{{\rm{s}},i+1}}{2}\left(-{\rm{\nabla }}{e}_{\mathrm{grav},i+\tfrac{1}{2}}\right.{\left.+{{\boldsymbol{f}}}_{\nu ,i+\tfrac{1}{2}}+{{\boldsymbol{f}}}_{\mathrm{ccor},{\rm{i}}+\tfrac{1}{2}}\right)}_{r,\theta ,\phi }.\end{array}\end{eqnarray} \tag{ 105 }$$

Given the time-averaged states of p and u to the left and right of each zone interface, ${\xi }_{i+\tfrac{1}{2}}$, the time-averaged values, ${p}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$ and ${u}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$, of the pressure and velocity of the zone interface itself are computed by connecting ${p}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$ and ${u}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$ to the time-averaged left and right states by the Rankine–Hugoniot relations; that is:

$$\begin{eqnarray}&&\displaystyle \frac{{p}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}-\langle p{\rangle }_{L,i+\tfrac{1}{2}}}{\langle W{\rangle }_{L,i+\tfrac{1}{2}}}+\left({u}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}-\langle u{\rangle }_{L,i+\tfrac{1}{2}}\right)=0,\end{eqnarray} \tag{ 106 }$$

$$\begin{eqnarray}\begin{array}{rcl}\langle W{\rangle }_{L,i+\tfrac{1}{2}}^{2} & = & {\left[\langle \gamma {\rangle }_{L}\langle p{\rangle }_{L}\langle \rho {\rangle }_{L}\right]}_{i+\tfrac{1}{2}}\,{\left[1+\displaystyle \frac{\langle \gamma {\rangle }_{L}+1}{2\langle \gamma {\rangle }_{L}}\left(\displaystyle \frac{{p}^{n+\tfrac{1}{2}}}{\langle p{\rangle }_{L}}-1\right)\right]}_{i+\tfrac{1}{2}},\end{array}\end{eqnarray} \tag{ 107 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{p}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}-\langle p{\rangle }_{R,i+\tfrac{1}{2}}}{\langle W{\rangle }_{R,i+\tfrac{1}{2}}}+\left({u}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}-\langle u{\rangle }_{R,i+\tfrac{1}{2}}\right)=0,\end{eqnarray} \tag{ 108 }$$

$$\begin{eqnarray}\begin{array}{rcl}\langle W{\rangle }_{R,i+\tfrac{1}{2}}^{2} & = & {\left[\langle \gamma {\rangle }_{R}\langle p{\rangle }_{R}\langle \rho {\rangle }_{R}\right]}_{i+\tfrac{1}{2}}\,{\left[1+\displaystyle \frac{\langle \gamma {\rangle }_{R}+1}{2\langle \gamma {\rangle }_{R}}\left(\displaystyle \frac{{p}^{n+\tfrac{1}{2}}}{\langle p{\rangle }_{R}}-1\right)\right]}_{i+\tfrac{1}{2}}.\end{array}\end{eqnarray} \tag{ 109 }$$

Equations (106)–(109) are iterated for ${p}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$ and ${u}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$ by the secant method.

Having determined ${p}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$ and ${u}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$ for each of the zone interfaces, the Lagrangian update proceeds as follows. With the values of ${u}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$ determined, the zone interfaces are considered impenetrable and their positions are updated by

$$\begin{eqnarray}\begin{array}{rclrcl}{\xi }_{i+\tfrac{1}{2}}^{n+1^{\prime} } & = & {\xi }_{i+\tfrac{1}{2}}^{n}+\displaystyle \frac{{u}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}\,{\rm{\Delta }}t}{{ \mathcal R }};\, & { \mathcal R } & = & \left\{\begin{array}{cc}1 & r{\rm{ \mbox{-} direction}},\\ {r}_{i} & \theta {\rm{ \mbox{-} direction}},\\ {r}_{i}\sin {\theta }_{i} & \phi {\rm{ \mbox{-} direction}},\end{array}\right.\end{array}\end{eqnarray} \tag{ 110 }$$

where the superscripts n and $n+1^{\prime}$ denote the value of a variable at time t and at the end of the Lagrangian step at time $t+{\rm{\Delta }}t$, respectively. A superscript n + $\tfrac{1}{2}$ denotes a time-centered value of a variable. We reserve the superscript $n+1$ for the value of a variable after both the Lagrangian and the remap step have been completed. From Equation (92), the density is then updated by

$$\begin{eqnarray}\begin{array}{rclrcl}{\rho }_{i}^{n+1^{\prime} } & = & {\rho }_{i}^{n}\,\displaystyle \frac{{V}_{i}^{n}}{{V}_{i}^{n+1^{\prime} }};\, & {V}_{i} & = & \left\{\begin{array}{cc}\tfrac{1}{3}\left({r}_{i+\tfrac{1}{2}}^{3}-{r}_{i-\tfrac{1}{2}}^{3}\right) & r{\rm{ \mbox{-} direction}},\\ {r}_{i}\left(\cos {\theta }_{i-\tfrac{1}{2}}-\cos {\theta }_{i+\tfrac{1}{2}}\right) & \theta {\rm{ \mbox{-} direction}},\\ {r}_{i}\sin {\theta }_{i}\left({\phi }_{i+\tfrac{1}{2}}-{\phi }_{i-\tfrac{1}{2}}\right) & \phi {\rm{ \mbox{-} direction}}.\end{array}\right.\end{array}\end{eqnarray} \tag{ 111 }$$

Because of the conservation of mass in each zone during the Lagrangian step, as expressed by Equation (92), and in differenced form by Equation (111), Equation (95) for the change in ${Y}_{{\rm{e}}}$ and Equation (96) for the change in the composition mass fractions, X_i, with their right-hand sides set to zero, state the obvious: ${Y}_{{\rm{e}}}$ and X_i are unchanged during the Lagrangian step.

Equation (93) with the help of Equation (92) can be rewritten in differential form as

$$\begin{eqnarray}&&\displaystyle \frac{d{\boldsymbol{u}}}{{dt}}=-\displaystyle \frac{1}{\rho }{\rm{\nabla }}p-{\rm{\nabla }}{e}_{\mathrm{grav}}+{{\boldsymbol{f}}}_{\nu }+{{\boldsymbol{f}}}_{\mathrm{ccor}},\end{eqnarray} \tag{ 112 }$$

where, again, the time derivative is a Lagrangian time derivative. In component form, Equation (112) becomes

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{{du}}_{r}}{{dt}} & = & -\displaystyle \frac{1}{\rho }\displaystyle \frac{\partial p}{\partial r}-\displaystyle \frac{\partial {e}_{\mathrm{grav}}}{\partial r}+{f}_{\nu ,r}+\displaystyle \frac{{u}_{\theta }^{2}+{u}_{\phi }^{2}}{r},\\ \displaystyle \frac{{{du}}_{\theta }}{{dt}} & = & -\displaystyle \frac{1}{\rho }\displaystyle \frac{1}{r}\displaystyle \frac{\partial p}{\partial \theta }-\displaystyle \frac{1}{r}\displaystyle \frac{\partial {e}_{\mathrm{grav}}}{\partial \theta }+{f}_{\nu ,\theta }+\displaystyle \frac{{u}_{\phi }^{2}}{r\sin \theta }\cos \theta -\displaystyle \frac{{u}_{r}{u}_{\theta }}{r},\\ \displaystyle \frac{{{du}}_{\phi }}{{dt}} & = & -\displaystyle \frac{1}{\rho }\displaystyle \frac{1}{r\sin \theta }\displaystyle \frac{\partial p}{\partial \phi }-\displaystyle \frac{1}{r\sin \theta }\displaystyle \frac{\partial {e}_{\mathrm{grav}}}{\partial \phi }+\,{f}_{\nu ,\phi }-\displaystyle \frac{{u}_{\theta }{u}_{\phi }}{r\sin \theta }\cos \theta -\displaystyle \frac{{u}_{r}{u}_{\phi }}{r}.\end{array}\end{eqnarray} \tag{ 113 }$$

For the radial sweep, the neutrino stress term, ${f}_{\nu ,r}$, is computed as described by the right-most term on the right-hand side of Equation (250). Because of the RbR approximation adopted by Chimera for neutrino transport, the values for ${f}_{\nu ,\theta }$ and ${f}_{\nu ,\phi }$ appearing in the θ- and ϕ-sweeps cannot be obtained directly as an outcome of the transport as can ${f}_{\nu ,r}$ for the radial sweep. To include ${f}_{\nu ,\theta }$ and ${f}_{\nu ,\phi }$ in an approximate way, we regard the matter as being completely neutrino-opaque at densities above 10¹² ${{\rm{g\; cm}}}^{-3}$ and completely transparent at lower densities. The neutrino distribution in each zone is thus assumed to behave like an isotropic, completely relativistic gas at densities above 10¹² ${{\rm{g\; cm}}}^{-3}$, whose effect on the hydrodynamics is computed by means of their corresponding pressure, ${p}_{\nu }$, and specific energy, ${e}_{\nu }$. For densities below 10¹² ${{\rm{g\; cm}}}^{-3}$, ${p}_{\nu }$ and ${e}_{\nu }$ are set to zero. The neutrino stress for the θ- and ϕ-sweeps is either that of an isotropic gas entrained with the matter (above 10¹² ${{\rm{g\; cm}}}^{-3}$) or zero (below 10¹² ${{\rm{g\; cm}}}^{-3}$). As a result, we set ${f}_{\nu ,\theta }$ and ${f}_{\nu ,\phi }$ to zero and include ${p}_{\nu }$ (nonzero above 10¹² ${{\rm{g\; cm}}}^{-3}$) in the material pressure, and the sum is PPM interpolated and incorporated into ${p}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$ and is then used to update the specific internal energy, with ${e}_{\nu ,i}$ incorporated into and later extracted from ${e}_{\mathrm{int},i}$.

The finite difference approximations to Equations (113) are

$$\begin{eqnarray}\begin{array}{rcll}{u}_{r,i}^{n+1^{\prime} } & = & {u}_{r,i}^{n}+\displaystyle \frac{\tfrac{1}{2}\left({A}_{i-\tfrac{1}{2}}^{n+\tfrac{1}{2}}+{A}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}\right)}{{V}_{i}^{n+\tfrac{1}{2}}}\displaystyle \frac{1}{{\rho }_{i}^{n+\tfrac{1}{2}}}\left({p}_{i-\tfrac{1}{2}}^{n+\tfrac{1}{2}}-{p}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}\right){\rm{\Delta }}t & +\,\left(-{{\rm{\nabla }}}_{r}{e}_{\mathrm{grav},i}^{n+\tfrac{1}{2}}+{f}_{r,\nu \,i}^{n}+{f}_{r,\mathrm{cenfugal},i}^{n+\tfrac{1}{2}}\right){\rm{\Delta }}t,\end{array}\end{eqnarray} \tag{ 114 }$$

$$\begin{eqnarray}\begin{array}{rcl}{u}_{\theta ,\phi ,i}^{n+1^{\prime} } & = & {u}_{\theta ,\phi ,i}^{n}+\displaystyle \frac{\tfrac{1}{2}\left({A}_{i-\tfrac{1}{2}}^{n+\tfrac{1}{2}}+{A}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}\right)}{{V}_{i}^{n+\tfrac{1}{2}}}\displaystyle \frac{1}{{\rho }_{i}^{n+\tfrac{1}{2}}}\,\left({\left(p+{p}_{\nu }\right)}_{i-\tfrac{1}{2}}^{n+\tfrac{1}{2}}-{\left(p+{p}_{\nu }\right)}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}\right){\rm{\Delta }}t\\ & & +\left(-{{\rm{\nabla }}}_{\theta ,\phi }\,{e}_{\mathrm{grav},i}^{n+\tfrac{1}{2}}+{f}_{\theta ,\phi ,\mathrm{cenfugal},i}^{n+\tfrac{1}{2}}\right){\rm{\Delta }}t,\end{array}\end{eqnarray} \tag{ 115 }$$

where ${A}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}$ is defined by $\delta {V}_{i+\tfrac{1}{2}}={A}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}{u}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}{\rm{\Delta }}t$. $\delta {V}_{i+\tfrac{1}{2}}$ is the volume swept out by the change in the position of the ith + 1 interface in the time interval ${\rm{\Delta }}t$ and is given by

$$\begin{eqnarray}\begin{array}{ll}{A}_{i+\tfrac{1}{2}} & =\,\left\{\begin{array}{cc}\tfrac{1}{3}\left({\left({r}_{i+\tfrac{1}{2}}^{n+1^{\prime} }\right)}^{2}+{r}_{i+\tfrac{1}{2}}^{n+1^{\prime} }{r}_{i+\tfrac{1}{2}}^{n}+{\left({r}_{i+\tfrac{1}{2}}^{n}\right)}^{2}\right) & r \mbox{-} \mathrm{direction},\\ \left(\cos {\theta }_{i+\tfrac{1}{2}}^{n}-\cos {\theta }_{i+\tfrac{1}{2}}^{n+1^{\prime} }\right)/\left({\theta }_{i+\tfrac{1}{2}}^{n+1^{\prime\prime} }-{\theta }_{i+\tfrac{1}{2}}^{n}\right) & \theta \mbox{-} \mathrm{direction},\\ 1 & \phi \mbox{-} \mathrm{direction}.\end{array}\right.\end{array}\end{eqnarray} \tag{ 116 }$$

${f}_{\mathrm{cenfugal},i}^{n+\tfrac{1}{2}}$ includes the last term in Equations (113) for the r-direction sweep and the second-to-last term in Equations (113) for the θ-direction sweep. The last term in Equation (113) for the θ-direction sweep is an expression for the change in ${u}_{\theta }$ due to a change in the radial position by virtue of angular momentum conservation. Rather than including this term in Equation (113), its contribution to the change in u_θ is updated in the r-sweep by the equivalent expression

$$\begin{eqnarray}&&{u}_{\theta ,i}^{n+1^{\prime} }={u}_{\theta ,i}^{n}\,\displaystyle \frac{{r}_{i}^{n}}{{r}_{i}^{n+1^{\prime} }}.\end{eqnarray} \tag{ 117 }$$

Similarly, the last two terms in Equation (113) for the ϕ-direction sweeps are expressions for angular momentum conservation about the z-axis due to a change in the θ and radial positions, respectively. Rather than including these terms in Equation (113), their contributions to the change in u_ϕ are updated in the θ- and r-direction sweeps by the equivalent expressions

$$\begin{eqnarray}&&{u}_{\phi ,i}^{n+1^{\prime} }={u}_{\phi ,i}^{n}\,\displaystyle \frac{{r}_{i}\sin {\theta }_{i}^{n}}{{r}_{i}\sin {\theta }_{i}^{n+1^{\prime} }},\qquad {u}_{\phi ,i}^{n+1^{\prime} }={u}_{\phi ,i}^{n}\,\displaystyle \frac{{r}_{i}^{n}}{{r}_{i}^{n+1^{\prime} }}.\end{eqnarray} \tag{ 118 }$$

The time-centered value of the gravitational potential, ${\phi }_{{\rm{g}}}$, is obtained by extrapolation, as described in Section 4.6. The two centrifugal force terms are differenced as follows:

$$\begin{eqnarray}&&{f}_{\mathrm{cenfugal},i}^{n+\tfrac{1}{2}}=\displaystyle \frac{{\left({u}_{\theta ,i}^{n+\tfrac{1}{2}}\right)}^{2}+{\left({u}_{\phi ,i}^{n+\tfrac{1}{2}}\right)}^{2}}{{r}_{i}^{n+\tfrac{1}{2}}}\ r{\rm{ \mbox{-} direction}},\end{eqnarray} \tag{ 119 }$$

$$\begin{eqnarray}&&{f}_{\mathrm{cenfugal},i}^{n+\tfrac{1}{2}}=\displaystyle \frac{{\left({u}_{\phi ,i}^{n+\tfrac{1}{2}}\right)}^{2}}{{r}_{i}^{n+\tfrac{1}{2}}\sin {\theta }_{i}^{n+\tfrac{1}{2}}}\cos {\theta }_{i}^{n+\tfrac{1}{2}}\ \theta {\rm{ \mbox{-} direction}},\end{eqnarray} \tag{ 120 }$$

where the time-centering of r_i in the r-direction and ${\theta }_{i}$ in the θ-direction is performed explicitly by averaging their values at time n and $n+1$, and the time-centering of the other variables is approximately accomplished by the symmetric way in which the directional splitting is performed, as described in Section 2.1. Finally, the neutrino stress term ${f}_{\nu ,i}^{n}$ is not time centered. A second execution of the neutrino transport would be required to center it. This term is small and slowly varying in time, so we include it to the first-order only and avoid the significant additional cost that would be paid were we to include it with second-order accuracy.

The specific internal energy is updated differently depending on whether the zone is in the vicinity of a shock or away from shocks. In the vicinity of a shock, the results of solving the Rankine–Hugoniot equations must be used, as the flow there is non-isentropic. In this case, the specific total energy is updated as given, in general form, by Equation (99) for the radial sweep, and Equation (94) for the θ- and ϕ-sweeps. Specifically, for the radial-sweep, the specific total energy, ${e}_{\mathrm{tot}}={e}_{\mathrm{int}}+{e}_{\mathrm{kin}}+{e}_{\mathrm{grav}}$, is updated by

$$\begin{eqnarray}\begin{array}{rcll}{e}_{\mathrm{tot},i}^{n+1^{\prime} } & = & {e}_{\mathrm{tot},i}^{n}-\displaystyle \frac{\left({A}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}{p}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}{u}_{r,i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}-{A}_{i-\tfrac{1}{2}}^{n+\tfrac{1}{2}}{p}_{i-\tfrac{1}{2}}^{n+\tfrac{1}{2}}{u}_{r,i-\tfrac{1}{2}}^{n+\tfrac{1}{2}}\right)}{{V}_{i}^{n+\tfrac{1}{2}}{\rho }_{i}^{n+\tfrac{1}{2}}}{\rm{\Delta }}t & +\,\displaystyle \frac{1}{2}\left({u}_{r,i}^{n}+{u}_{r,i}^{n+1^{\prime} }\right){f}_{\nu ,i}^{n}{\rm{\Delta }}t\quad r{\rm{ \mbox{-} direction}},\end{array}\end{eqnarray} \tag{ 121 }$$

where ${u}_{i}^{n+1^{\prime} }$ is given by Equation (114). The $\partial {e}_{\mathrm{grav}}/\partial t$ term in Equation (99) has been omitted at this stage but is included later in the radial sweep. For the θ- and ϕ-sweeps, the specific energy, ${e}_{\mathrm{tot}^{\prime} }={e}_{\mathrm{int}}+{e}_{\mathrm{kin}}$, is updated by

$$\begin{eqnarray}&&\begin{array}{l}{\left({e}_{\mathrm{tot}^{\prime} }+{e}_{\nu }\right)}_{i}^{n+1^{\prime} }={\left({e}_{\mathrm{tot}^{\prime} }+{e}_{\nu }\right)}_{i}^{n}\\ \,-\,\displaystyle \frac{\left({A}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}{\left(p+{p}_{\nu }\right)}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}{u}_{i+\tfrac{1}{2}}^{n+\tfrac{1}{2}}-{A}_{i-\tfrac{1}{2}}^{n+\tfrac{1}{2}}{\left(p+{p}_{\nu }\right)}_{i-\tfrac{1}{2}}^{n+\tfrac{1}{2}}\,{u}_{i-\tfrac{1}{2}}^{n+\tfrac{1}{2}}\right)}{{V}_{i}^{n+\tfrac{1}{2}}{\rho }_{i}^{n+\tfrac{1}{2}}}\ {\rm{\Delta }}t\\ \,+\,\displaystyle \frac{1}{2}\left({u}_{i}^{n}+{u}_{i}^{n+1^{\prime} }\right)\left(-{\rm{\nabla }}{e}_{\mathrm{grav},i}^{n+\tfrac{1}{2}}\right){\rm{\Delta }}t\quad \theta {\rm{ \mbox{-} }},\ \phi \mbox{-} {\rm{directions}},\end{array}\end{eqnarray} \tag{ 122 }$$

followed by

$$\begin{eqnarray}\begin{array}{rcl}{e}_{\mathrm{tot}^{\prime} }^{n+1^{\prime} } & = & ({e}_{\mathrm{tot}^{\prime} }+{e}_{\nu }{)}_{i}^{n+1^{\prime} }-{e}_{\nu }^{n+1^{\prime} }\,=\,({e}_{\mathrm{tot}^{\prime} }+{e}_{\nu }{)}_{i}^{n+1^{\prime} }-{e}_{\nu }^{n}{\left(\displaystyle \frac{{\rho }^{n+1^{\prime} }}{{\rho }^{n}}\right)}^{4/3}.\end{array}\end{eqnarray} \tag{ 123 }$$

Away from shocks, the specific energy could still be updated as above, but errors might then arise during the remap step when subtracting the specific kinetic energy from the specific total energy. The problem arises from the use of $\langle {\boldsymbol{u}}{\rangle }^{2}$ in the expression for the specific kinetic energy rather than $\langle {{\boldsymbol{u}}}^{2}\rangle$. The two expressions can differ importantly in supersonic flow and near reflection boundaries where the gradient of ${\boldsymbol{u}}$ can be large (see Blondin & Lufkin 1993, for a discussion of this point). Computing $\langle {{\boldsymbol{u}}}^{2}\rangle$ would be costly in a multidimensional simulation, as it would involve a multidimensional integration over the components of ${\boldsymbol{u}}$ and may not be well defined. Instead, Chimera updates the specific internal energy using the first law of thermodynamics, assuming isentropic and constant-composition flow, as non-isentropic changes due to nuclear and neutrino sources are computed elsewhere by operator splitting.

The update of the specific internal energy thus takes the form

$$\begin{eqnarray}&&{e}_{\mathrm{int},i}^{n+1^{\prime} }={e}_{\mathrm{int},i}^{n}+\displaystyle \frac{{p}_{i}^{n+\tfrac{1}{2}}}{{\left({\rho }_{i}^{n+\tfrac{1}{2}}\right)}^{2}}\left({\rho }_{i}^{n+1^{\prime} }-{\rho }_{i}^{n}\right),\end{eqnarray} \tag{ 124 }$$

where the time-centering of the pressure p is accomplished by a predictor-corrector loop, completing the Lagrangian step.

Following the Lagrangian step, in the case of the θ- and ϕ-sweeps, the grid is remapped back to the configuration that prevailed before the Lagrangian step, thus, making the combination of a Lagrangian step and a remap step effectively an Eulerian step. In the case of the radial sweep, the grid is remapped back to a configuration specified by the regridder, which will be described in Section 4.5. Via the regridder options, the user can specify that the grid following the Lagrangian step be left as is, remapped back to the configuration that prevailed before the Lagrangian step, or, by invoking one of the regridder algorithms, remapped to a configuration that tends to optimize the resolution of structures that develop during a simulation.

4.4.1. Remapping Mass, Momenta, and Angular Momenta

For quantities like the mass, specific momenta (momenta per gram), and specific angular momenta, the remapping procedure is straightforward. Denoting, as before, the values of the grid and other variables after the Lagrangian step by the superscript $n+1^{\prime}$, and after the remap step by the superscript $n+1$, the difference $\delta \xi$, between the values of the grid variables at $n+1^{\prime}$ and $n+1$, given by

$$\begin{eqnarray}&&\delta {\xi }_{i+\tfrac{1}{2}}={\xi }_{i+\tfrac{1}{2}}^{n+1^{\prime} }-{\xi }_{i+\tfrac{1}{2}}^{n+1}\end{eqnarray} \tag{ 125 }$$

is computed, as is the volume $\delta {V}_{i+\tfrac{1}{2}}$ contained within $\delta {\xi }_{i+\tfrac{1}{2}}$. The latter is given by

$$\begin{eqnarray}\delta {V}_{i+\tfrac{1}{2}}=\left\{\begin{array}{cc}\displaystyle \frac{1}{3}\left[{\left({\xi }_{i-\tfrac{1}{2}}^{n+1^{\prime} }\right)}^{3}-{\left({\xi }_{i+\tfrac{1}{2}}^{n+1}\right)}^{3}\right] & r{\rm{ \mbox{-} direction}},\\ {r}_{i}\left(\cos {\xi }_{i-\tfrac{1}{2}}^{n+1}-\cos {\xi }_{i+\tfrac{1}{2}}^{n+1^{\prime} }\right) & \theta {\rm{ \mbox{-} direction}},\\ {r}_{i}\sin {\theta }_{i}\left({\phi }_{i+\tfrac{1}{2}}^{n+1^{\prime} }-{\phi }_{i+\tfrac{1}{2}}^{n+1}\right) & \phi {\rm{ \mbox{-} direction}}.\end{array}\right.\end{eqnarray} \tag{ 126 }$$

If $\delta {\xi }_{i+\tfrac{1}{2}}\gt 0$, the grid interface ${\xi }_{i+\tfrac{1}{2}}^{n+1}$ is placed by the remap step between ${\xi }_{i-\tfrac{1}{2}}^{n+1^{\prime} }$ and ${\xi }_{i+\tfrac{1}{2}}^{n+1^{\prime} }$. The mass advected across the zone interface ${\xi }_{i+\tfrac{1}{2}}$ is next computed by first constructing a PPM profile, $\rho (\xi )$, of the density ${\rho }_{i}^{n+1^{\prime} }$. The value, $\langle \rho {\rangle }_{L,i+\tfrac{1}{2}}^{n+1^{\prime} }$, of the interpolated density $\rho (\xi )$ averaged over the interval $\delta {\xi }_{i+\tfrac{1}{2}}$ to the left of interface ${\xi }_{i+\tfrac{1}{2}}^{n+1^{\prime} }$ is then computed by Equation (102), with $\delta {\xi }_{i+\tfrac{1}{2}}$ replacing ${c}_{{\rm{s}},i}{\rm{\Delta }}t$. The mass advected is then given by

$$\begin{eqnarray}&&\delta {{ \mathcal M }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }=\langle \rho {\rangle }_{L,i+\tfrac{1}{2}}^{n+1^{\prime} }\delta {V}_{i+\tfrac{1}{2}}.\end{eqnarray} \tag{ 127 }$$

The zone mass after the remap step, ${{ \mathcal M }}_{i}^{n+1}$, is then computed from the the zone mass before the remap step, ${{ \mathcal M }}_{i}^{n+1^{\prime} }$, and the advected mass $\delta {{ \mathcal M }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }$ by

$$\begin{eqnarray}&&{{ \mathcal M }}_{i}^{n+1}={{ \mathcal M }}_{i}^{n+1^{\prime} }-\delta {{ \mathcal M }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }+\delta {{ \mathcal M }}_{i-\tfrac{1}{2}}^{n+1^{\prime} },\end{eqnarray} \tag{ 128 }$$

and the density ${\rho }_{i}^{n+1}$ is then computed by

$$\begin{eqnarray}&&{\rho }_{i}^{n+1}=\displaystyle \frac{{{ \mathcal M }}_{i}^{n+1}}{{V}_{i}^{n+1}}.\end{eqnarray} \tag{ 129 }$$

Having determined the advected mass $\delta {{ \mathcal M }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }$, the remapping of any specific (i.e., per gram) quantity ${a}_{i}^{n+1^{\prime} }$ proceeds by determining the amount, $\delta {{ \mathcal A }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }$, of that quantity advected. As in the case of the density, a piecewise parabolic interpolation profile, $a(\xi )$, of ${a}_{i}^{n+1^{\prime} }$ is constructed and the average $\langle a{\rangle }_{L,i+\tfrac{1}{2}}^{n+1^{\prime} }$ of that quantity over the interval $\delta {\xi }_{i+\tfrac{1}{2}}$ is computed using Equation (102). The mass of $a(\xi )$ contained within $\delta {\xi }_{i+\tfrac{1}{2}}$ is finally computed as the penultimate step by

$$\begin{eqnarray}&&\delta {{ \mathcal A }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }=\langle a{\rangle }_{L,i+\tfrac{1}{2}}^{n+1^{\prime} }\delta {{ \mathcal M }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }.\end{eqnarray} \tag{ 130 }$$

If $\delta {\xi }_{i+\tfrac{1}{2}}\lt 0$, the grid interface ${\xi }_{i+\tfrac{1}{2}}^{n+1}$ is placed by the remap step between ${\xi }_{i+\tfrac{1}{2}}^{n+1^{\prime} }$ and ${\xi }_{i+\tfrac{3}{2}}^{n+1^{\prime} }$, and the quantity of $a(\xi )$ within $\delta {\xi }_{i+\tfrac{1}{2}}$ is advected from the right to the left of interface ${\xi }_{i+\tfrac{1}{2}}^{n+1}$. The procedure is the same as described above for the $\delta {\xi }_{i+\tfrac{1}{2}}\gt 0$ case, except that $\langle \rho {\rangle }_{R,i+\tfrac{1}{2}}^{n+1^{\prime} }$ and $\langle a{\rangle }_{R,i+\tfrac{1}{2}}^{n+1^{\prime} }$ are computed from ${\rho }_{i+1}^{n+1^{\prime} }$ and ${a}_{i+1}^{n+1^{\prime} }$ by Equation (103), giving $\delta {{ \mathcal M }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }$ and $\delta {{ \mathcal A }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }$ to the right of interface ${\xi }_{i+\tfrac{1}{2}}^{n+1^{\prime} }$, with $\delta {\xi }_{i+\tfrac{1}{2}}$ replacing ${c}_{{\rm{s}},i}{\rm{\Delta }}t$, as before, but now with a negative value of $\delta {V}_{i+\tfrac{1}{2}}$. Therefore, the quantity $\delta {{ \mathcal M }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }$ computed from $\langle \rho {\rangle }_{R,i+\tfrac{1}{2}}^{n+1^{\prime} }$ by an equation analogous to Equation (127), and the quantity $\delta {{ \mathcal A }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }$ computed from $\langle a{\rangle }_{R,i+\tfrac{1}{2}}^{n+1^{\prime} }$ by an equations analogous to Equation (130), are both negative.

The remap step is finally completed by performing the advection:

$$\begin{eqnarray}&&{a}_{i}^{n+1}=\displaystyle \frac{{a}_{i}^{n+1^{\prime} }{{ \mathcal M }}_{i}^{n+1^{\prime} }-\delta {{ \mathcal A }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }+\delta {{ \mathcal A }}_{i-\tfrac{1}{2}}^{n+1^{\prime} }}{{{ \mathcal M }}_{i}^{n+1}}.\end{eqnarray} \tag{ 131 }$$

Negative values of $\delta {{ \mathcal A }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }$ or $\delta {{ \mathcal A }}_{i-\tfrac{1}{2}}^{n+1^{\prime} }$ mean simply that the advection proceeds from right to left rather than in the other direction. The advection step is conservative since the same amount of a given quantity enters a zone as leaves the adjacent zone. Again, the sign of $\delta {{ \mathcal A }}_{i+\tfrac{1}{2}}^{n+1^{\prime} }$ determines whether the advection is from zone i to zone $i+1$, or vice versa.

4.4.2. Remapping Composition and Electron Fraction

4.4.3. Multiple EoSs and the Energy Remap

Chimera is designed to accommodate multiple EoSs that are applied in contiguous density ranges. As explained above in Section 3.2, for example, Chimera has used the LS EoS at densities above 10¹¹ ${{\rm{g\; cm}}}^{-3}$, and a different EoS below that density. Since Chimera updates the specific energy directly rather than the temperature and uses the updated specific energy and other needed thermodynamic variables to update the temperature, slight differences in the energy zeros at the boundary between two EoS's could result in unphysical temperature updates. Specific energy differences between two EoSs could also arise from peculiarities or approximations peculiar to each EoS. In either case, we will refer to the potential difference in specific energies given by two EoS's for the same thermodynamic state as a zero-energy offset. Unphysical temperature updates could happen, for example, during the remap step if matter from a zone linked to one EoS is advected into an adjacent zone linked to a different EoS. The quantity of energy advected would contain the difference in the energy zeros as well as the physically relevant energy. This problem could affect the radial sweep but not the θ- or ϕ-sweeps as the same EoS is always used along a θ- or ϕ-directed ray.

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To avoid this problem, Chimera overlaps by four radial zones in either direction the specific internal energy at the boundary between EoSs as shown in Figure 13, where radial zones i − 1 and lower are tied to a particular EoS A, while zones i and higher are tied to a different EoS B. The two EoSs have different zero-energy offsets, as indicated by their vertical displacement, and an overlap of four zones in either direction. This overlap enables PPM profiles of the specific internal energy for zones i − 1 and below to be constructed using EoS A up to the interface i − $\tfrac{1}{2}$. Likewise, PPM profiles of the specific internal energy for zones i and above can be constructed using EoS B. The example in Figure 13 is one in which the zone interface ${\xi }_{i-\tfrac{1}{2}}$ is remapped a distance $\delta {\xi }_{i-\tfrac{1}{2}}$ to the left of its original position, its new position being indicated by the vertical red dashed line. This entails that a quantity of specific internal energy contained within $\delta {\xi }_{i-\tfrac{1}{2}}$ be advected from the left to the right of zone interface ${\xi }_{i-\tfrac{1}{2}}$. Chimera performs this advection by advecting the energy within $\delta {\xi }_{i-\tfrac{1}{2}}$ as given by EoS A out of zone i − 1, and advecting the energy within the same $\delta {\xi }_{i-\tfrac{1}{2}}$ but as given by EoS B into zone i. Since the specific internal energy advected out of a zone and into the adjacent zone is consistent with the different EoSs tied to each of the two zones, the unphysical temperature jump that would occur if the zero-energy offset had not been accounted for in the advection is avoided.

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4.4.4. Nuclear Binding Energy

In advecting the specific energy between two adjacent zones during a remap, the specific internal energy is split into a nuclear binding energy component, ${e}_{\mathrm{bind}}$, and the rest of the energy, and the two components are advected separately. For example, the specific internal energy, ${e}_{\mathrm{int}}$, is split into ${e}_{\mathrm{bind}}$ and ${e}_{\mathrm{th}}={e}_{\mathrm{int}}-{e}_{\mathrm{bind}}$, and remapped as follows:

$$\begin{eqnarray}&&{e}_{\mathrm{int},i}^{n+1}=\displaystyle \frac{\left({e}_{\mathrm{th},i}^{n+1^{\prime} }+{e}_{\mathrm{bind},i}^{n+1^{\prime} }\right)\delta {{ \mathcal M }}_{i}^{n+1^{\prime} }-\delta {E}_{\mathrm{th},i+\tfrac{1}{2}}^{n+1^{\prime} }-\delta {E}_{\mathrm{bind},i+\tfrac{1}{2}}^{n+1^{\prime} }+\delta {E}_{\mathrm{th},i-\tfrac{1}{2}}^{n+1^{\prime} }+\delta {E}_{\mathrm{bind},i-\tfrac{1}{2}}^{n+1^{\prime} }}{\delta {{ \mathcal M }}_{i}^{n+1}},\end{eqnarray} \tag{ 132 }$$

where ${e}_{\mathrm{th},i}^{n+1^{\prime} }$ and ${e}_{\mathrm{bind},i}^{n+1^{\prime} }$ are the specific internal energy minus the specific binding energy, and the specific binding energy, respectively, of zone i, and $\delta {E}_{\mathrm{th},i\pm \tfrac{1}{2}}^{n+1^{\prime} }$ and $\delta {E}_{\mathrm{bind},i\pm \tfrac{1}{2}}^{n+1^{\prime} }$ are the internal energy minus the binding energy and the binding energy, respectively, transferred through outer zone edge $\left(i+\tfrac{1}{2}\right)$ and inner zone edge $\left(i-\tfrac{1}{2}\right)$ of zone i. The masses $\delta {{ \mathcal M }}_{i}^{n+1^{\prime} }$ and $\delta {{ \mathcal M }}_{i}^{n+1}$ are the masses of zone i before and after the remap, respectively, as given by Equation (127). This mode of energy advection is appropriate for advecting non-NSE material during a remap, as the non-NSE material being advected does not necessarily have the same composition and, therefore, binding energy, as the original material in the zone from which it is being advected. The composition of the material being advected is obtained by integrating over the PPM profile constructed for each ionic mass fraction, and then normalizing the sum to unity. Since the PPM profiles of different ionic mass fractions may be differently shaped, the composition that results after integrating over the portion of the profiles being advected and normalizing can result in a composition and binding energy different from the original. In the energy advection procedure described above, the binding energy of the advecting material is computed once its composition is ascertained, and the rest of the advected energy, $\delta {e}_{\mathrm{th}}$, is also computed by integrating the PPM profile of ${e}_{\mathrm{th}}={e}_{\mathrm{int}}-{e}_{\mathrm{bind}}$ over the advecting mass. The net energy advected should thus reflect both the correct nuclear binding energy of the advecting material, as well as its thermal component.

4.4.5. Energy Remap for the θ- and ϕ-sweeps and the Preliminary Remap for the Radial Sweep

The energy remap described above is the ultimate step in the θ- and ϕ-sweep hydrodynamics, and the penultimate step in the radial sweep hydrodynamics. Away from shocks, the specific internal energy is remapped as given by Equation (132), as opposed to remapping the sum of the specific internal plus specific kinetic energy, ${e}_{\mathrm{tot}^{\prime} ,i}={e}_{\mathrm{int}}+{e}_{\mathrm{kin}}$. This is permissible as the flow is isentropic apart from the contributions of nuclear and neutrino source terms, which have been included elsewhere in the radial sweep (see the text above Equation (124) for the motivation for this approach). In the vicinity of shocks, the specific energy, ${e}_{\mathrm{tot}^{\prime} }$, which has been evolved during the Lagrangian step using the Rankine–Hugoniot equations, must be remapped, and the specific internal energy is extracted afterwards. We define ${e}_{\mathrm{tot}^{\prime\prime} }={e}_{\mathrm{tot}^{\prime} }-{e}_{\mathrm{bind}}={e}_{\mathrm{th}}+{e}_{\mathrm{kin}}$ and remap ${e}_{\mathrm{tot}^{\prime\prime} }$ and ${e}_{\mathrm{bind}}$ separately, as described above. To use consistent values for the left and right states for calculating a PPM profile of ${e}_{\mathrm{tot}^{\prime\prime} }$ (i.e., consistent with the velocity remap), we define

$$\begin{eqnarray}\begin{array}{rcl}{e}_{\mathrm{tot}^{\prime\prime} ,L,i}^{n+1^{\prime} } & = & {e}_{\mathrm{th},L,i}^{n+1^{\prime} }+\displaystyle \frac{1}{2}\left[{\left({u}_{r,L,i}^{n+1^{\prime} }\right)}^{2}+{\left({u}_{\theta ,L,i}^{n+1^{\prime} }\right)}^{2}+{\left({u}_{\phi ,L,i}^{n+1^{\prime} }\right)}^{2}\right],\\ {e}_{\mathrm{tot}^{\prime\prime} ,R,i}^{n+1^{\prime} } & = & {e}_{\mathrm{th},R,i}^{n+1^{\prime} }+\displaystyle \frac{1}{2}\left[{\left({u}_{r,R,i}^{n+1^{\prime} }\right)}^{2}+{\left({u}_{\theta ,R,i}^{n+1^{\prime} }\right)}^{2}+{\left({u}_{\phi ,R,i}^{n+1^{\prime} }\right)}^{2}\right].\end{array}\end{eqnarray} \tag{ 133 }$$

Having calculated the PPM profile of ${e}_{\mathrm{tot}^{\prime\prime} }$, the quantity of ${e}_{\mathrm{tot}^{\prime\prime} }$ advected across a given zone interface is computed by Equations (127) or (130), and the remapping of ${e}_{\mathrm{tot}^{\prime\prime} }$ is performed by an equation analogous to Equation (132). After the remapping of ${e}_{\mathrm{tot}^{\prime\prime} }$ and ${e}_{\mathrm{bind}}$, the specific internal energy is extracted from their sum, ${e}_{\mathrm{tot}^{\prime} }$.

4.4.6. Recomputation of the Gravitational Potential and the Computation of $\partial {e}_{\mathrm{grav}}/\partial t$

Following the remap in the radial-sweep of the mass, momenta, angular momenta, the independent thermodynamic variables, and the preliminary remap of the specific energy, the specific gravitational potential, ${e}_{\mathrm{grav}}^{n+1}$, is computed. In the case in which the spherically symmetric component of the gravitational potential is computed by means of a general relativistic approximation, the remapped pressure, specific energy, and neutrino contributions are used as sources of gravity as well as the density, which necessitated the preliminary remap of the specific energy. It is at this stage of the radial-sweep that the quantity $\partial {e}_{\mathrm{grav}}/\partial t$ is computed and added, in accordance with Equation (99), to the specific total energy, ${e}_{\mathrm{tot}}^{n+1^{\prime} }$, that was computed by Equation (121) during the Lagrangian step. The most direct procedure for calculating $\partial {e}_{\mathrm{grav}}/\partial t$ would be to calculate the gravitational potential, ${e}_{\mathrm{grav}}^{n+1^{\prime} }$, after the Lagrangian step, interpolate the initial gravitational potential, ${e}_{\mathrm{grav}}^{n}$, to the Lagrangian grid to get the quantity ${e}_{\mathrm{grav},I-L}^{n}$, and approximate $\partial {e}_{\mathrm{grav}}/\partial t$ by $\left({e}_{\mathrm{grav}}^{n+1^{\prime} }-{e}_{\mathrm{grav},I-L}^{n}\right)/{\rm{\Delta }}t$. This would work well for one-dimensional simulations, but for multidimensional simulations, a given radial grid edge, ${\xi }_{i+\tfrac{1}{2}}$, after the Lagrangian step is a function of θ and/or ϕ, making the gravitational potential difficult to compute at this point. Instead, the gravitational potential is computed after the remap of the radial grid and interpolated as a function of θ and/or ϕ to the Lagrangian grid, thereby obtaining ${e}_{\mathrm{grav},F-L}^{n+1}$. The time derivative of the gravitational potential added to ${e}_{\mathrm{tot}}^{n+1^{\prime} }$ is thus given, as a function of θ and/or ϕ, by

$$\begin{eqnarray}&&\displaystyle \frac{\partial {e}_{\mathrm{grav}}}{\partial t}=\displaystyle \frac{{e}_{\mathrm{grav},F-L}^{n+1}-{e}_{\mathrm{grav},I-L}^{n}}{{\rm{\Delta }}t}.\end{eqnarray} \tag{ 134 }$$

4.4.7. Final Radial-sweep Remap of the Total Energy

The final remapping of the specific total energy $\left({e}_{\mathrm{kin}}+{e}_{\mathrm{int}}+{e}_{\mathrm{grav}}\right)$ in the radial sweep begins with the specific energy, ${e}_{\mathrm{tot}^{\prime\prime} ,i}^{n+1^{\prime} }$, given by

$$\begin{eqnarray}&&{e}_{\mathrm{tot}^{\prime\prime} ,i}^{n+1^{\prime} }={e}_{\mathrm{tot},i}^{n+1^{\prime} }-{e}_{\mathrm{bind},i}^{n+1^{\prime} }-{e}_{\mathrm{kin},i}^{n+1^{\prime} }+{\left(\displaystyle \frac{\partial {e}_{\mathrm{grav}}}{\partial t}\right)}_{i}^{n+\tfrac{1}{2}}{\rm{\Delta }}t,\end{eqnarray} \tag{ 135 }$$

where ${e}_{\mathrm{tot},i}$ has been updated during the Lagrangian step, as given by Equation (121). Consistent left and right states of ${e}_{\mathrm{tot}^{\prime\prime} ,i}^{n+1^{\prime} }$ are determined as specified by Equations (133), PPM profiles of ${e}_{\mathrm{tot}^{\prime} ,i}^{n+1^{\prime} }={e}_{\mathrm{tot}^{\prime\prime} ,i}^{n+1^{\prime} }+{e}_{\mathrm{kin},i}^{n+1^{\prime} }$ are then obtained, and the amount of ${e}_{\mathrm{tot}^{\prime} \,i}^{n+1^{\prime} }$ to be advected across the zone interface are given by Equation (130) and the procedure outlined in the discussion below it. Remapping then proceeds in accordance with an equation analogous to Equation (132), and the final specific internal energy is extracted from ${e}_{\mathrm{tot},i}^{n+1}={e}_{\mathrm{tot}^{\prime} ,i}^{n+1}+{e}_{\mathrm{bind},i}^{n+1}$ by

$$\begin{eqnarray}&&{e}_{\mathrm{int},i}^{n+1}={e}_{\mathrm{tot},i}^{n+1}-{e}_{\mathrm{kin},i}^{n+1}-{e}_{\mathrm{grav},i}^{n+1}.\end{eqnarray} \tag{ 136 }$$

This completes the remap step of the radial-sweep hydrodynamics.

4.4.8. Suppression of Carbuncles

When shocks are aligned with one of the coordinate directions in multidimensional simulations, they are susceptible to an "odd–even decoupling" or "carbuncle" instability (Quirk 1994; Liou 2000; Sutherland et al. 2003). This can lead to a strong rippling of the shock front, which could, in turn, excite hydrodynamic instabilities in the post-shock region. Our scheme for suppressing this instability is the use of a "local oscillation filter" similar in philosophy to that described by Sutherland et al. (2003). This approach is local and does not affect the well-resolved features of the flow elsewhere. To suppress the carbuncle instability in the radial direction, which is where this instability typically arises in a supernova simulation, the angular (θ) and azimuthal (ϕ) remap steps are each followed by an examination of the radial velocities along the angular and azimuthal rays to search for radial velocity extrema. If there are at least three radial velocity extrema in any group of five adjacent zones, and if a shock is present, these zones are marked for "smoothing." For the particular case in which zones m and $m+1$ are marked for smoothing, the flux $\delta {{ \mathcal A }}_{m+\tfrac{1}{2}}$ of the quantity a, defined by

$$\begin{eqnarray}&&\delta {{ \mathcal A }}_{m+\tfrac{1}{2}}={c}_{\mathrm{smooth}}\left({a}_{m+1}-{a}_{m}\right)\min ({{ \mathcal M }}_{m},{{ \mathcal M }}_{m+1}),\end{eqnarray} \tag{ 137 }$$

is computed, where ${{ \mathcal M }}_{m}$ is the mass of zone m, and ${c}_{\mathrm{smooth}}$ is an empirical parameter. Experimentation has shown that a value of 0.075 for ${c}_{\mathrm{smooth}}$ works well. The final step in the procedure is to sweep across the angular and azimuthal rays and exchange the flux $\delta {{ \mathcal A }}_{m+\tfrac{1}{2}}$ between the zones marked for smoothing in a step analogous to that described by Equation (131). This reduces the difference in the values of the quantity a between adjacent zones, thereby inhibiting the growth of this difference. Applying this procedure to the quantities u_r and ${u}_{\phi }$, and to ${L}_{\phi }$ (to smooth ${u}_{\theta }$), proved sufficiently robust to suppress the carbuncle instability.

The PPM Lagrangian-Remap format permits the grid after the Lagrangian step to be remapped to a grid other than the initial grid from which the Lagrangian step originated. While the θ- and ϕ-grids are remapped back to their initial grids after the Lagrangian step, making them effectively Eulerian, Chimera uses the remapping freedom to provide the user with a number of remapping options for the radial grid to ensure that the grid continues to resolve important structures that arise during the course of a simulation. One option is for the radial grid to be remapped back to the initial grid after the Lagrangian step, making the grid effectively Eulerian as it is for the θ and ϕ grids. Another option is for the remapped grid to follow the mean motion of the fluid, referred to here and below as pseudo-Lagrangian, making the grid purely Lagrangian in the case of spherically symmetric fluid flow.

Currently, a number of more sophisticated options are available specific to the pre-bounce or post-bounce phase of a CCSN simulation. For both the pre-bounce phase and the post-bounce phase, an inner-outer boundary dividing the radial grid into an inner and an outer section is determined based on a number of user selected criteria. These criteria can differ between the pre-bounce and the post-bounce phases and can differ at user-selected time intervals during the post-bounce phase. During both the pre-bounce and the post-bounce phases, the outer grid can be selected to be pseudo-Lagrangian, which is useful if there are sharp chemical discontinuities in non-NSE material that need to be preserved, or Eulerian if advection through the outer boundary of a prescribed distribution of material is important. During the pre-bounce phase, the inner grid starts out as pseudo-Lagrangian, but blends into another grid between two user-selected densities. This second grid is constructed so that adjacent zones satisfy ${\rm{\Delta }}{r}_{i+1}={\rm{constant}}\times \ {\rm{\Delta }}{r}_{i}$, referred to here as a "zoomed grid," with the properties that the width of the outer zone of this zoomed grid is equal to the zone width of the first zone of the outer grid, and the inner zone tends to a user selected width when that zone reaches $3\times {10}^{14}$ ${{\rm{g\; cm}}}^{-3}$. The result is a smooth and smoothly evolving grid that can be tuned to provide the desired grid resolution at the proto-neutron star surface when it forms. During the post-bounce phase, the inner grid remains a zoomed grid from the core center to a density of 10¹⁴ ${{\rm{g\; cm}}}^{-3}$, with the central zone width such that it would attain a user-selected zone width at a density of $3\times {10}^{14}$ ${{\rm{g\; cm}}}^{-3}$. A second zoomed grid covers the density range from 10¹⁴ to 10¹² ${{\rm{g\; cm}}}^{-3}$, and a third covers the density range from 10¹² to 10¹⁰ ${{\rm{g\; cm}}}^{-3}$. Both of these latter two grids have the same number of zones, which are equal to a user-selected value. This ensures there are a sufficient number of zones to resolve the neutrinosphere as the proto-neutron star shrinks and as the density cliff forms near its edge. Finally, a fourth zoomed grid covers the region from the density of 10¹⁰ ${{\rm{g\; cm}}}^{-3}$ to the outer edge of the inner grid. The result of this regridding is again a smooth and smoothly evolving user-controlled grid designed to resolve the critical features that arise during the course of a CCSN simulation.

Self gravity can be chosen to be either one- or multi-dimensional, with a further choice of either a Newtonian or an approximate general relativistic monopole component. The approximate general relativistic gravitational potential used for the monopole component in the latter case is a modified Tolman–Oppenheimer–Volkoff (TOV) potential suggested by Marek et al. (2006, Case A) and described briefly below in Section 4.6.1. Multidimensional gravity is obtained by expanding the Newtonian gravitational potential in a multipole expansion as described by Müller & Steinmetz (1995) and below in Section 4.6.2. Approximate general relativistic multidimensional gravity is obtained by replacing the Newtonian monopole in the multipole expansion by the approximate general relativistic monopole.

4.6.1. One-dimensional Gravitational Potential

Newtonian monopole gravity is trivial. The radial zone-edged and zone-centered gravitational accelerations, ${g}_{i+\tfrac{1}{2}}$ and g_i, respectively, are given by

$$\begin{eqnarray}&&{g}_{i+\tfrac{1}{2}}=-{{GM}}_{i+\tfrac{1}{2}}/{R}_{i+\tfrac{1}{2}}^{2},\quad {g}_{i}=-{{GM}}_{i}/{R}_{i}^{2},\end{eqnarray} \tag{ 138 }$$

where ${M}_{i+\tfrac{1}{2}}$ is the rest mass enclosed in a volume of radius ${R}_{i+\tfrac{1}{2}}$, R_i the mass-averaged zone-centered radius, and G the gravitational constant. The radial zone-edged and zone-centered gravitational potentials, ${e}_{\mathrm{grav},i+\tfrac{1}{2}}$ and ${e}_{\mathrm{grav},i}$, respectively, are given by

$$\begin{eqnarray}\begin{array}{rclrcl}{e}_{\mathrm{grav},i-\tfrac{1}{2}} & = & {e}_{\mathrm{grav},i+\tfrac{1}{2}}-{g}_{i}{\rm{\Delta }}{R}_{i}, & \,{e}_{\mathrm{grav},i} & = & {e}_{\mathrm{grav},i+1}-{g}_{i+\tfrac{1}{2}}{\rm{\Delta }}{R}_{i+\tfrac{1}{2}},\end{array}\end{eqnarray} \tag{ 139 }$$

where at the outer edge of the radial grid

$$\begin{eqnarray}&&{e}_{\mathrm{grav},I+\tfrac{1}{2}}=-\displaystyle \frac{{{GM}}_{I+\tfrac{1}{2}}}{{R}_{I+\tfrac{1}{2}}};\quad {e}_{\mathrm{grav},I}={e}_{\mathrm{grav},I+\tfrac{1}{2}}-{g}_{I}\left({R}_{I+\tfrac{1}{2}}-{R}_{I}\right).\end{eqnarray} \tag{ 140 }$$

Approximate GR monopole gravity is computed by first iterating the following two equations (Marek et al. 2006, Case A) for ${M}_{\mathrm{TOV}}$:

$$\begin{eqnarray}\begin{array}{rcll}{M}_{\mathrm{TOV},I+\displaystyle \frac{1}{2}} & = & {M}_{\mathrm{TOV},I-\displaystyle \frac{1}{2}}+{{\rm{\Gamma }}}_{\mathrm{TOV},i}{\rm{\Delta }}{V}_{i} & \,\left[{\rho }_{i}+\displaystyle \frac{{\rho }_{i}\left({e}_{\mathrm{int},i}+{e}_{\nu ,i}\right)+{u}_{r,i}{{ \mathcal F }}_{\nu ,\mathrm{flux}}/({c}^{2}{{\rm{\Gamma }}}_{\mathrm{TOV},i})}{{c}^{2}}\right],\end{array}\end{eqnarray} \tag{ 141 }$$

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{TOV},i}=\sqrt{1+\displaystyle \frac{{u}_{r,i}^{2}-2{{GM}}_{\mathrm{TOV},i}/{R}_{i}}{{c}^{2}}},\end{eqnarray} \tag{ 142 }$$

where ${e}_{\mathrm{int},i}$ and ${e}_{\nu ,i}$ are the specific energy densities of matter and neutrinos, respectively, and ${u}_{r,i}$ is the radial velocity. With ${M}_{\mathrm{TOV},I}$ computed as $\tfrac{1}{2}\left({M}_{\mathrm{TOV},I-\tfrac{1}{2}}+{M}_{\mathrm{TOV},I+\tfrac{1}{2}}\right)$, the zone-centered gravitational force is computed by

$$\begin{eqnarray}\begin{array}{rcll}{g}_{i} & = & -G\displaystyle \frac{{M}_{\mathrm{TOV},I}+4\pi {R}_{i}^{3}\left({p}_{\mathrm{gas},i}+{p}_{\nu ,i}\right)/{c}^{2}}{{R}_{i}^{2}} & \,\displaystyle \frac{1+\left({e}_{\mathrm{int},i}+{p}_{\mathrm{gas},i}/{\rho }_{i}\right)/{c}^{2}}{{{\rm{\Gamma }}}_{\mathrm{TOV},i}^{2}},\end{array}\end{eqnarray} \tag{ 143 }$$

where ${p}_{\mathrm{gas},i}$ and ${p}_{\nu ,i}$ are the matter pressure and spherically averaged neutrino pressure, respectively. Once g_i is computed, ${g}_{i+\tfrac{1}{2}}$ is computed as $\left({g}_{i}+{g}_{i+1}\right)/2$, with ${g}_{0+\tfrac{1}{2}}=0$ and ${g}_{I+\tfrac{1}{2}}$ extrapolated from g_I and ${g}_{I-\tfrac{1}{2}}$. The zone-edged gravitational potential, ${e}_{\mathrm{grav},I+\tfrac{1}{2}}$, is then computed by Equation (139), and the zone-centered gravitational potential is given by ${g}_{i}\,=\left({g}_{i-\tfrac{1}{2}}+{g}_{i+\tfrac{1}{2}}\right)/2$.

4.6.2. Multipole Expansion of the Gravitational Potential—Axisymmetry

To incorporate nonspherical gravity, Chimera uses a scheme based on the method described by Müller & Steinmetz (1995) of expanding the integral Newtonian Poisson equation in a multipole expansion. When implementing approximate general relativistic gravity, the Newtonian monopole is replaced with the approximate general relativistic monopole (Marek et al. 2006, Case A) described above. Multipole gravity is implemented in both axisymmetric and three-dimensional simulations.

For axisymmetric simulations, this scheme utilizes the identity

$$\begin{eqnarray}&&\displaystyle \frac{1}{({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )}=\sum _{{\ell }=0}^{\infty }\displaystyle \frac{{r}_{\lt }^{{\ell }}}{{r}_{\gt }^{{\ell }+1}}{P}_{{\ell }}(\cos (\theta ^{\prime} -\theta ),\end{eqnarray} \tag{ 144 }$$

where

$$\begin{eqnarray}&&\displaystyle \frac{{r}_{\lt }^{{\ell }}}{{r}_{\gt }^{{\ell }+1}}={\rm{\Theta }}(r-r^{\prime} )\displaystyle \frac{r{{\prime} }^{{\ell }}}{{r}^{{\ell }+1}}+{\rm{\Theta }}(r^{\prime} -r)\displaystyle \frac{{r}^{{\ell }}}{r{{\prime} }^{{\ell }+1}},\end{eqnarray} \tag{ 145 }$$

${\rm{\Theta }}(x)$ is the Heaviside function, and P_ℓ is the Legendre polynomial of order ${\ell }$, to expand the Poisson integral in a Legendre series:

$$\begin{eqnarray}\begin{array}{rcl}{e}_{\mathrm{grav}}(r,\theta ) & = & -G{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{\rho ({\boldsymbol{r}}^{\prime} ){dV}^{\prime} }{({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )}\\ & = & -G{\displaystyle \int }_{0}^{\infty }r{{\prime} }^{2}{dr}^{\prime} {\displaystyle \int }_{-1}^{1}d\cos \left(\theta ^{\prime} -\theta \right){\displaystyle \int }_{0}^{2\pi }d\phi ^{\prime} \rho ({\boldsymbol{r}}^{\prime} )\displaystyle \sum _{{\ell }=0}^{\infty }\displaystyle \frac{{r}_{\lt }^{{\ell }}}{{r}_{\gt }^{{\ell }+1}}\\ & & \times \,\left({P}_{{\ell }}(\cos \theta ^{\prime} ){P}_{{\ell }}(\cos \theta )+2\displaystyle \sum _{m=1}^{{\ell }}\displaystyle \frac{({\ell }-m)!}{({\ell }+m)!}\right.\,\left.{P}_{{\ell }}^{m}(\cos \theta ^{\prime} ){P}_{{\ell }}^{m}(\cos \theta )\cos m(\phi ^{\prime} -\phi ,),\Space{0ex}{3.25ex}{0ex}\right)\\ & = & -2\pi G\displaystyle \sum _{{\ell }=0}^{\infty }\,{P}_{{\ell }}(\cos \theta ){\displaystyle \int }_{0}^{\infty }r{{\prime} }^{2}{dr}^{\prime} {\displaystyle \int }_{-1}^{1}d\cos \left(\theta ^{\prime} \right)\rho ({\boldsymbol{r}}^{\prime} )\displaystyle \frac{{r}_{\lt }^{{\ell }}}{{r}_{\gt }^{{\ell }+1}}{P}_{{\ell }}(\cos \theta ^{\prime} ).\end{array}\end{eqnarray} \tag{ 146 }$$

Here, the ${P}_{{\ell }}^{m}$ are the associated Legendre functions and the integral over $\phi ^{\prime}$ has been performed utilizing the assumption of axisymmetry. In differencing Equation (146), the spherical coordinate system utilized by Chimera enables the radius $r^{\prime}$ to be integrated over each spherical shell and the potential to be computed at zone interfaces. Given the singular nature of the Poisson equation, this avoids the problem of the gravitational self-interaction, which can lead to a nonconvergence of the multipole expansion, as pointed out by Couch et al. (2013). Equation (146) in differenced form then becomes

$$\begin{eqnarray}\begin{array}{rcl}{e}_{\mathrm{grav},i+\displaystyle \frac{1}{2},j+\displaystyle \frac{1}{2}} & = & -2\pi {{GP}}_{{\ell }j+\tfrac{1}{2}}\left[\displaystyle \sum _{{\ell }=0}^{{N}_{{\ell }}}\displaystyle \sum _{i^{\prime} =1}^{i}\displaystyle \frac{{r}_{i^{\prime} +\tfrac{1}{2}}^{{\ell }+3}-{r}_{i^{\prime} -\tfrac{1}{2}}^{{\ell }+3}}{({\ell }+3){r}_{i+\tfrac{1}{2}}^{{\ell }+1}}{{ \mathcal P }}_{{\ell },i^{\prime} }\right.\\ & & +\,\displaystyle \sum _{{\ell }=1,({\ell }\ne 2)}^{{N}_{{\ell }}}\displaystyle \sum _{i^{\prime} =i}^{I-1}\displaystyle \frac{{r}_{i^{\prime} +\tfrac{3}{2}}^{2-{\ell }}-{r}_{i^{\prime} +\tfrac{1}{2}}^{2-{\ell }}}{2-{\ell }}{r}_{i+\tfrac{1}{2}}^{{\ell }}{{ \mathcal P }}_{{\ell },i^{\prime} +1} & +\left.\,\displaystyle \sum _{i^{\prime} =i}^{I-1}{r}_{i+\tfrac{1}{2}}^{2}\mathrm{ln}\displaystyle \frac{{r}_{i^{\prime} +\tfrac{3}{2}}}{{r}_{i^{\prime} +\tfrac{1}{2}}}{{ \mathcal P }}_{2,i^{\prime} +1}\right]\quad i\geqslant 1,\end{array}\end{eqnarray} \tag{ 147 }$$

$$\begin{eqnarray}\begin{array}{rcl}{e}_{\mathrm{grav},\displaystyle \frac{1}{2},j+\displaystyle \frac{1}{2}} & = & -2\pi G\left[\displaystyle \sum _{i^{\prime} =i}^{I-1}\displaystyle \frac{{r}_{i^{\prime} +\tfrac{3}{2}}^{2}-{r}_{i^{\prime} +\tfrac{1}{2}}^{2}}{2}{{ \mathcal P }}_{0i^{\prime} +1}+\displaystyle \frac{1}{2}{r}_{\tfrac{3}{2}}^{2}{{ \mathcal P }}_{\mathrm{0,1}}\right]\,i=0,\end{array}\end{eqnarray} \tag{ 148 }$$

where

$$\begin{eqnarray}\begin{array}{rclrcl}{{ \mathcal P }}_{{\ell },i} & = & \displaystyle \sum _{j=1}^{J}{P}_{\mathrm{int},j+\displaystyle \frac{1}{2}}{\rho }_{i,j},\quad \mathrm{and}\quad {P}_{\mathrm{int},j+\displaystyle \frac{1}{2}} & & = & {\displaystyle \int }_{\cos {\theta }_{j-\tfrac{1}{2}}}^{\cos {\theta }_{j+\tfrac{1}{2}}}{P}_{{\ell }}(\cos \theta )d(\cos \theta ).\end{array}\end{eqnarray} \tag{ 149 }$$

The Legendre polynomials up to the specified order and the angular integrations of these polynomials over the angular zone widths are generated as an initialization step. The Legendre polynomials are first computed at the angular zone edges using

$$\begin{eqnarray}&&{P}_{0,j+\tfrac{1}{2}}=1,\quad {P}_{1,j+\tfrac{1}{2}}=\cos \left({\theta }_{j+\tfrac{1}{2}}\right),\end{eqnarray} \tag{ 150 }$$

and the recurrence relation for ℓ > 1:

$$\begin{eqnarray}&&{P}_{{\ell },j+\tfrac{1}{2}}=\displaystyle \frac{2{\ell }-1}{{\ell }}\cos \left({\theta }_{j+\tfrac{1}{2}}\right){P}_{{\ell }-1,j+\tfrac{1}{2}}-\displaystyle \frac{{\ell }-1}{{\ell }}{P}_{{\ell }-2,j+\tfrac{1}{2}}.\end{eqnarray} \tag{ 151 }$$

The Legendre polynomials are then integrated over the zone widths, using

$$\begin{eqnarray}&&{P}_{\mathrm{int},0,j}\equiv {\displaystyle \int }_{\cos \left({\theta }_{j-\tfrac{1}{2}}\right)}^{\cos \left({\theta }_{j+\tfrac{1}{2}}\right)}{P}_{0}(y){dy}={P}_{1,j+\tfrac{1}{2}}-{P}_{1,j-\tfrac{1}{2}},\end{eqnarray} \tag{ 152 }$$

and then

$$\begin{eqnarray}\begin{array}{rclcl}{P}_{\mathrm{int},{\ell },j} & \equiv & {\displaystyle \int }_{\cos \left({\theta }_{j-\tfrac{1}{2}}\right)}^{\cos \left({\theta }_{j+\tfrac{1}{2}}\right)}{P}_{{\ell }}(y){dy} & = & \displaystyle \frac{{P}_{{\ell }-1,j+\tfrac{1}{2}}-{P}_{{\ell }-1,j-\tfrac{1}{2}}}{2{\ell }+1}-\displaystyle \frac{{P}_{{\ell }+1,j+\tfrac{1}{2}}-{P}_{{\ell }+1,j-\tfrac{1}{2}}}{2{\ell }+1}.\end{array}\end{eqnarray} \tag{ 153 }$$

To assess the accuracy of the gravitational potential expansion as a function of the number of multipoles used, and for code verification, the gravitational potentials computed by Equations (147) and (148) for a Maclaurin spheroid are compared to those given by an exact analytic solution in Section 5.7.

4.6.3. Multipole Expansion of the Gravitational Potential—Non-axisymmetry

For non-axisymmetric simulations, this scheme utilizes the identity

$$\begin{eqnarray}&&\displaystyle \frac{1}{({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )}=\sum _{{\ell }=0}^{\infty }\sum _{m=-{\ell }}^{{\ell }}\displaystyle \frac{4\pi }{2{\ell }+1}\displaystyle \frac{{r}_{\lt }^{{\ell }}}{{r}_{\gt }^{{\ell }+1}}{Y}_{{\ell }}^{m}(\theta ,\phi ){Y}_{{\ell }}^{m* }(\theta ^{\prime} ,\phi ^{\prime} ),\end{eqnarray} \tag{ 154 }$$

to expand the gravitational potential in spherical harmonics, where ${Y}_{{\ell }}^{m* }$ denotes the complex conjugate of the spherical harmonic ${Y}_{{\ell }}^{m}$, and ${r}_{\lt }^{{\ell }}/{r}_{\gt }^{{\ell }+1}$ is defined by Equation (145). The expansion is given by

$$\begin{eqnarray}\begin{array}{rcl}{e}_{\mathrm{grav}}(r,\theta ,\phi ) & = & -G{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{\rho ({\boldsymbol{r}}^{\prime} ){dV}^{\prime} }{({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )},\\ & = & -G\displaystyle \sum _{{\ell }=0}^{\infty }\displaystyle \frac{4\pi }{2{\ell }+1}\displaystyle \sum _{m=-{\ell }}^{{\ell }}{Y}_{{\ell }}^{m}(\theta ,\phi ){\displaystyle \int }_{0}^{\infty }{\left(r^{\prime} \right)}^{2}\,{dr}^{\prime} \displaystyle \frac{{r}_{\lt }^{{\ell }}}{{r}_{\gt }^{{\ell }+1}}{\displaystyle \int }_{4\pi }d{\rm{\Omega }}^{\prime} {Y}_{{\ell }}^{m* }(\theta ^{\prime} ,\phi ^{\prime} )\rho (r^{\prime} ,\theta ^{\prime} ,\phi ^{\prime} ),\\ & = & -G\displaystyle \sum _{{\ell }=0}^{\infty }\displaystyle \frac{4\pi }{2{\ell }+1}\displaystyle \sum _{m=-{\ell }}^{{\ell }}{Y}_{{\ell }}^{m}(\theta ,\phi )\left(\displaystyle \frac{1}{{r}^{{\ell }+1}}{C}_{(r)}^{{\ell }m}+{r}^{{\ell }}{D}_{(r)}^{{\ell }m}\right),\end{array}\end{eqnarray} \tag{ 155 }$$

where

$$\begin{eqnarray}\begin{array}{rclcl}{C}_{(r)}^{{\ell }m} & = & {\displaystyle \int }_{4\pi }d{\rm{\Omega }}^{\prime} \,{Y}_{{\ell }}^{m* }(\theta ^{\prime} ,\phi ^{\prime} ){\displaystyle \int }_{0}^{r}{dr}^{\prime} {\left(r^{\prime} \right)}^{{\ell }+2}\rho (r^{\prime} ,\theta ^{\prime} ,\phi ^{\prime} ) & = & \displaystyle \frac{1}{\sqrt{4\pi }}{\displaystyle \int }_{0}^{r}{dr}^{\prime} {\left(r^{\prime} \right)}^{{\ell }+2}A(r^{\prime} ,{\ell },m),\end{array}\end{eqnarray} \tag{ 156 }$$

$$\begin{eqnarray}\begin{array}{rclrcl}{D}_{(r)}^{{\ell }m} & = & {\displaystyle \int }_{4\pi }d{\rm{\Omega }}^{\prime} \,{Y}_{{\ell }}^{m* }(\theta ^{\prime} ,\phi ^{\prime} ){\displaystyle \int }_{r}^{\infty }{dr}^{\prime} {\left(r^{\prime} \right)}^{1-{\ell }}\rho (r^{\prime} ,\theta ^{\prime} ,\phi ^{\prime} ) & & = & \displaystyle \frac{1}{\sqrt{4\pi }}{\displaystyle \int }_{r}^{\infty }{dr}^{\prime} {\left(r^{\prime} \right)}^{1-{\ell }}A(r^{\prime} ,{\ell },m),\end{array}\end{eqnarray} \tag{ 157 }$$

and where

$$\begin{eqnarray}\begin{array}{rcll}A(r^{\prime} ,{\ell },m) & = & {\displaystyle \int }_{0}^{2\pi }d\phi ^{\prime} {\displaystyle \int }_{0}^{\pi }d\theta ^{\prime} \sin \theta ^{\prime} {H}_{(m)}{\tilde{P}}_{{\ell }}^{m} & (\cos \theta ^{\prime} ){e}^{-{im}\phi ^{\prime} }\rho (r^{\prime} ,\theta ^{\prime} .\phi ^{\prime} ),\end{array}\end{eqnarray} \tag{ 158 }$$

and

$$\begin{eqnarray}{H}_{(m)}=\left\{\begin{array}{cl}1 & m=0\\ \displaystyle \frac{1}{\sqrt{2}} & m\ne 0\end{array}\right..\end{eqnarray} \tag{ 159 }$$

The occasional use of $i=\sqrt{-1}$ here should not cause confusion with the radial index i. The spherical harmonics ${Y}_{{\ell }}^{m}(\theta ,\phi )$ have been written as functions of the normalized associated Legendre functions ${\tilde{P}}_{{\ell }}^{m}(\cos \theta )$:

$$\begin{eqnarray}&&{Y}_{{\ell }}^{m}(\theta ,\phi )=\displaystyle \frac{1}{\sqrt{4\pi }}{\tilde{P}}_{{\ell }}^{m}\cos \theta ){e}^{{im}\phi }{H}_{(m)},\end{eqnarray} \tag{ 160 }$$

and the latter are given in terms of the unnormalized associated Legendre polynomials ${P}_{{\ell }}^{m}(\cos \theta )$ by

$$\begin{eqnarray}{\tilde{P}}_{{\ell }}^{m}(\cos \theta )={P}_{{\ell }}^{m}(\cos \theta )\times \left\{\begin{array}{cc}\sqrt{2{\ell }+1} & m=0\\ \sqrt{\displaystyle \frac{2(2{\ell }+1)({\ell }-m)!}{2{\ell }+1}} & m\ne 0\end{array}\right.,\end{eqnarray} \tag{ 161 }$$

where ${\tilde{P}}_{{\ell }}^{m}(\cos \theta )$ satisfies the relation

$$\begin{eqnarray}&&{\tilde{P}}_{{\ell }}^{-m}(\cos \theta )={\left(-1\right)}^{m}{\tilde{P}}_{{\ell }}^{m}(\cos \theta ).\end{eqnarray} \tag{ 162 }$$

The normalized associated Legendre functions, ${\tilde{P}}_{{\ell }}^{m}(\cos \theta )$, and their integrals, ${P}_{\mathrm{int}}({\ell },m,j)$, defined below by Equation (172) are calculated during an initial setup step by using the subroutines developed by NGA¹⁰ based on the algorithms of Paul (1978) and Gerstl (1980).

To transform $A(r^{\prime} ,{\ell },m)$ into a form suitable for calculation, we first change variables from $\theta ^{\prime}$ to $y^{\prime} \equiv \cos \theta ^{\prime}$, to obtain

$$\begin{eqnarray}\begin{array}{rcl}A(r^{\prime} ,{\ell },m) & = & {H}_{(m)}{\displaystyle \int }_{0}^{2\pi }d\phi ^{\prime} {\displaystyle \int }_{-1}^{1}{dy}^{\prime} {\tilde{P}}_{{\ell }}^{m}(y^{\prime} ){e}^{-{im}\phi ^{\prime} }\rho (r^{\prime} ,y^{\prime} .\phi ^{\prime} ),\end{array}\end{eqnarray} \tag{ 163 }$$

which, separating real and imaginary parts, can be written as

$$\begin{eqnarray}&&A(r^{\prime} ,{\ell },m)={A}_{r}(r^{\prime} ,{\ell },m)+{{iA}}_{i}(r^{\prime} ,{\ell },m),\end{eqnarray} \tag{ 164 }$$

where

$$\begin{eqnarray}\begin{array}{rcll}{A}_{r}(r^{\prime} ,{\ell },m) & = & {H}_{(m)}{\displaystyle \int }_{0}^{2\pi }d\phi ^{\prime} {\displaystyle \int }_{-1}^{1}{dy}^{\prime} {\tilde{P}}_{{\ell }}^{m}(y^{\prime} ) & \cos \left(m\phi ^{\prime} \right)\rho (r^{\prime} ,y^{\prime} ,\phi ^{\prime} ),\end{array}\end{eqnarray} \tag{ 165 }$$

$$\begin{eqnarray}\begin{array}{rcll}{A}_{i}(r^{\prime} ,{\ell },m) & = & -{H}_{(m)}{\displaystyle \int }_{0}^{2\pi }d\phi ^{\prime} {\displaystyle \int }_{-1}^{1}{dy}^{\prime} {\tilde{P}}_{{\ell }}^{m}(y^{\prime} ) & \sin \left(m\phi ^{\prime} \right)\rho (r^{\prime} ,y^{\prime} ,\phi ^{\prime} ).\end{array}\end{eqnarray} \tag{ 166 }$$

Next, we define the variables

$$\begin{eqnarray}&&{z}_{1}^{{\prime} }\equiv \sin \left(m\phi ^{\prime} \right),\quad {z}_{2}^{{\prime} }\equiv \cos \left(m\phi ^{\prime} \right),\end{eqnarray} \tag{ 167 }$$

and substitute them into Equations (165) and (166) to obtain

$$\begin{eqnarray}&&{A}_{r}(r^{\prime} ,{\ell },m)=\displaystyle \frac{{H}_{(m)}}{m}\int {{dz}}_{1}^{{\prime} }{\displaystyle \int }_{-1}^{1}{dy}^{\prime} {\tilde{P}}_{{\ell }}^{m}(y^{\prime} )\rho (r^{\prime} ,y^{\prime} ,{z}_{1}^{{\prime} }),\end{eqnarray} \tag{ 168 }$$

$$\begin{eqnarray}&&{A}_{i}(r^{\prime} ,{\ell },m)=\displaystyle \frac{{H}_{(m)}}{m}\int {{dz}}_{2}^{{\prime} }{\displaystyle \int }_{-1}^{1}{dy}^{\prime} {\tilde{P}}_{{\ell }}^{m}(y^{\prime} )\rho (r^{\prime} ,y^{\prime} ,{z}_{2}^{{\prime} }).\end{eqnarray} \tag{ 169 }$$

Note the symmetry conditions

$$\begin{eqnarray}\begin{array}{rclrcl}{A}_{r}(r^{\prime} ,{\ell },-m) & = & {\left(-1\right)}^{m}{A}_{r}(r^{\prime} ,{\ell },m),\, & {A}_{i}(r^{\prime} ,{\ell },-m) & = & {\left(-1\right)}^{m+1}{A}_{i}(r^{\prime} ,{\ell },m).\end{array}\end{eqnarray} \tag{ 170 }$$

To discretize the ${A}_{r}(r^{\prime} ,{\ell },m)$, we introduce a j index, $j=1,\cdots ,J$, and a k index, k = 1, ⋯, K, to denote the zone centers of the angular variable, θ, and the azimuthal variable, ϕ, respectively, analogous to the index i = 1, ⋯, I, which is illustrated in Figure 12 and used in this section for the radial index. Half-integer values of i, j, and k refer to zone edges. Primed and unprimed indices will refer to source and field quantities, respectively. We have

$$\begin{eqnarray}&&{A}_{r}(r^{\prime} ,{\ell },m)=\sum _{k^{\prime} =1}^{{N}_{k}}\sum _{j^{\prime} =1}^{{N}_{j}}{S}_{\mathrm{int}}(m,k^{\prime} ){P}_{\mathrm{int}}({\ell },m,j^{\prime} )\rho (r^{\prime} ,j^{\prime} ,k^{\prime} ),\end{eqnarray} \tag{ 171 }$$

where

$$\begin{eqnarray}&&{P}_{\mathrm{int}}({\ell },m,j^{\prime} )\equiv {\displaystyle \int }_{{y}_{j^{\prime} -\tfrac{1}{2}}}^{{y}_{j^{\prime} +\tfrac{1}{2}}}{\tilde{P}}_{{\ell }}^{m}(y^{\prime} ){dy}^{\prime} ,\end{eqnarray} \tag{ 172 }$$

and where

$$\begin{eqnarray}{S}_{\mathrm{int}}(m,k^{\prime} )\equiv \left\{\begin{array}{cc}\displaystyle \frac{1}{m\sqrt{2}}\left[\sin \left(m{\phi }_{k^{\prime} +\tfrac{1}{2}}\right)-\sin \left(m{\phi }_{k^{\prime} -\tfrac{1}{2}}\right)\right] & m\ne 0\\ {\phi }_{k^{\prime} +\tfrac{1}{2}}-{\phi }_{k^{\prime} -\tfrac{1}{2}} & m=0\end{array}\right..\end{eqnarray} \tag{ 173 }$$

Similarly,

$$\begin{eqnarray}&&{A}_{i}(r^{\prime} ,{\ell },m)=\sum _{k^{\prime} =1}^{{N}_{k}}\sum _{j^{\prime} =1}^{{N}_{j}}{C}_{\mathrm{int}}(m,k^{\prime} ){P}_{\mathrm{int}}({\ell },m,j^{\prime} )\rho (r^{\prime} ,j^{\prime} ,k^{\prime} ),\end{eqnarray} \tag{ 174 }$$

where

$$\begin{eqnarray}{C}_{\mathrm{int}}(m,k^{\prime} )\equiv \left\{\begin{array}{cc}\tfrac{1}{m\sqrt{2}}\left[\cos \left(m{\phi }_{k^{\prime} +\tfrac{1}{2}}\right)-\cos \left(m{\phi }_{k^{\prime} -\tfrac{1}{2}}\right)\right] & m\ne 0\\ 0 & m=0\end{array}\right..\end{eqnarray} \tag{ 175 }$$

To construct the gravitational potential, we have, from Equations (155)–(157) and (160),

$$\begin{eqnarray}\begin{array}{rcl}{e}_{\mathrm{grav}}(r,\theta ,\phi ) & = & -G\displaystyle \sum _{{\ell }=0}^{\infty }\displaystyle \frac{4\pi }{2{\ell }+1}\displaystyle \sum _{m=-{\ell }}^{{\ell }}\displaystyle \frac{{H}_{(m)}}{\sqrt{4\pi }}{\tilde{P}}_{{\ell }}^{m}(\cos \theta ){e}^{{im}\phi }\left(\displaystyle \frac{1}{{r}^{{\ell }+1}}{C}_{(r)}^{{\ell }m}+{r}^{{\ell }}{D}_{(r)}^{{\ell }m}\right),\\ & = & -G\displaystyle \sum _{{\ell }=0}^{\infty }\displaystyle \frac{4\pi }{2{\ell }+1}\displaystyle \sum _{m=-{\ell }}^{{\ell }}{H}_{(m)}{\tilde{P}}_{{\ell }}^{m}(\cos \theta )\\ & & \times \left[\displaystyle \frac{{e}^{{im}\phi }}{{r}^{{\ell }+1}}{\displaystyle \int }_{0}^{r}{dr}^{\prime} {\left(r^{\prime} \right)}^{{\ell }+2}A(r^{\prime} ,{\ell },m)\right.\left.+{e}^{{im}\phi }{e}^{{\ell }}{\displaystyle \int }_{r}^{\infty }{dr}^{\prime} {\left(r^{\prime} \right)}^{1-{\ell }}A(r^{\prime} ,{\ell },m)\right].\end{array}\end{eqnarray} \tag{ 176 }$$

Now, the expression ${\tilde{P}}_{{\ell }}^{m}(\cos \theta ){e}^{{im}\phi }A(r^{\prime} ,{\ell },m)$ in Equation (176) can be written

$$\begin{eqnarray}&&\begin{array}{l}{\tilde{P}}_{{\ell }}^{m}(\cos \theta ){e}^{{im}\phi }A(r^{\prime} ,{\ell },m)=\,{\tilde{P}}_{{\ell }}^{m}(\cos \theta )\left[\cos \left(m\phi \right)A(r^{\prime} ,{\ell },m)+i\sin \left(m\phi \right)A(r^{\prime} ,{\ell },m)\right]\\ \,\,=\,{\tilde{P}}_{{\ell }}^{m}(\cos \theta )\left[{B}_{r}(r^{\prime} ,{\ell },m)+{{iB}}_{i}(r^{\prime} ,{\ell },m)\right],\end{array}\end{eqnarray} \tag{ 177 }$$

where the real and imaginary parts of $B(r^{\prime} ,{\ell },m)$ are given by

$$\begin{eqnarray}\begin{array}{rcll}{B}_{r}(r^{\prime} ,{\ell },m,\phi ) & = & \cos \left(m\phi \right){A}_{r}(r^{\prime} ,{\ell },m) & -\sin \left(m\phi \right){A}_{i}(r^{\prime} ,{\ell },m),\end{array}\end{eqnarray} \tag{ 178 }$$

$$\begin{eqnarray}\begin{array}{rcll}{B}_{i}(r^{\prime} ,{\ell },m,\phi ) & = & \sin \left(m\phi \right){A}_{r}(r^{\prime} ,{\ell },m) & +\cos \left(m\phi \right){A}_{i}(r^{\prime} ,{\ell },m).\end{array}\end{eqnarray} \tag{ 179 }$$

Using the symmetry conditions expressed by Equations (162) and (170), we have

$$\begin{eqnarray}&&{\tilde{P}}_{{\ell }}^{-m}(\cos \theta ){B}_{r}(r^{\prime} ,{\ell },-m,\phi )={\tilde{P}}_{{\ell }}^{m}(\cos \theta ){B}_{r}(r^{\prime} ,{\ell },m,\phi ),\end{eqnarray} \tag{ 180 }$$

and

$$\begin{eqnarray}&&{\tilde{P}}_{{\ell }}^{-m}(\cos \theta ){B}_{i}(r^{\prime} ,{\ell },-m,\phi )=-{\tilde{P}}_{{\ell }}^{m}(\cos \theta ){B}_{i}(r^{\prime} ,{\ell },m,\phi ),\end{eqnarray} \tag{ 181 }$$

which gives

$$\begin{eqnarray}\begin{array}{ll}{\tilde{P}}_{{\ell }}^{-m}(\cos \theta ){B}_{r}(r^{\prime} ,{\ell },-m,\phi )+{\tilde{P}}_{{\ell }}^{m}(\cos \theta ){B}_{r}(r^{\prime} ,{\ell },m,\phi ) & =\,2{\tilde{P}}_{{\ell }}^{m}(\cos \theta ){B}_{r}(r^{\prime} ,{\ell },m,\phi ),\end{array}\end{eqnarray} \tag{ 182 }$$

and

$$\begin{eqnarray}\begin{array}{ll}{\tilde{P}}_{{\ell }}^{-m}(\cos \theta ){B}_{i}(r^{\prime} ,{\ell },-m,\phi ) & +\,{\tilde{P}}_{{\ell }}^{m}(\cos \theta ){B}_{i}(r^{\prime} ,{\ell },m,\phi )=0.\end{array}\end{eqnarray} \tag{ 183 }$$

Thus, the imaginary part of Equation (177) cancels out when summed over negative and positive m, and Equation (182) for the real part of $B(r^{\prime} ,{\ell },-m,\phi )$ can be used to replace the summation of m from $-{\ell }$ to ${\ell }$ to a summation from zero to ℓ.

The construction of the gravitational potential in Chimera begins with the computation of $A(r^{\prime} ,{\ell },m)$, defined in Equation (158) and discretized as described by Equations (171)–(175). With ${P}_{\mathrm{int}}$, S_int, and C_int given by Equations (172), (173), and (175), respectively, and computed in the initial setup, $A(r^{\prime} ,{\ell },m)$ is computed from

$$\begin{eqnarray}\begin{array}{rcl}A(i^{\prime} ,{\ell },m) & = & \displaystyle \sum _{k^{\prime} =1}^{K}\sum _{j^{\prime} =1}^{J}\left[{S}_{\mathrm{int}}(m,k^{\prime} ){P}_{\mathrm{int}}({\ell },m,j^{\prime} ){\rho }_{i^{\prime} ,j^{\prime} ,k^{\prime} }\right.\left.\,+{{iC}}_{\mathrm{int}}(m,k^{\prime} ){P}_{\mathrm{int}}({\ell },m,j^{\prime} ){\rho }_{i^{\prime} ,j^{\prime} ,k^{\prime} }\right].\end{array}\end{eqnarray} \tag{ 184 }$$

Using Equations (182) and (183), the quantities ϕ_in(r) and ϕ_out(r), given, respectively, by ${e}^{{im}\phi }{C}_{(r)}^{{\ell }m}$ and ${e}^{{im}\phi }{D}_{(r)}^{{\ell }m}$ (Equations (156) and (157)), are then computed recursively by

$$\begin{eqnarray}\begin{array}{rcl}{\phi }_{\mathrm{in}\displaystyle \frac{1}{2},k+\displaystyle \frac{1}{2}}({\ell },m) & = & 0,\\ {\phi }_{\mathrm{in}i+\displaystyle \frac{1}{2},k+\displaystyle \frac{1}{2}}({\ell },m) & = & {\phi }_{\mathrm{in}i-\displaystyle \frac{1}{2},k+\displaystyle \frac{1}{2}}({\ell },m){\left(\displaystyle \frac{{r}_{i+\tfrac{1}{2}}}{{r}_{i+\tfrac{3}{2}}}\right)}^{{\ell }+1} & +{B}_{r}(r^{\prime} ,{\ell },m,{\phi }_{k+\tfrac{1}{2}})\displaystyle \frac{{r}_{i+\tfrac{1}{2}}^{2}-{r}_{i-\tfrac{1}{2}}^{2}{\left(\tfrac{{r}_{i+\tfrac{1}{2}}}{{r}_{i+\tfrac{3}{2}}}\right)}^{{\ell }+1}}{{\ell }+3},\end{array}\end{eqnarray} \tag{ 185 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\phi }_{\mathrm{out}I+\displaystyle \frac{1}{2},k+\displaystyle \frac{1}{2}}({\ell },m) & = & 0,\\ {\phi }_{\mathrm{out}i+\displaystyle \frac{1}{2},k+\displaystyle \frac{1}{2}}({\ell },m) & = & {\phi }_{\mathrm{out}i+\displaystyle \frac{3}{2},k+\displaystyle \frac{1}{2}}({\ell },m){\left(\displaystyle \frac{{r}_{i+\tfrac{3}{2}}}{{r}_{i+\tfrac{5}{2}}}\right)}^{{\ell }}+{B}_{r}\left(r^{\prime} ,{\ell },m,{\phi }_{k+\tfrac{1}{2}}\right)\displaystyle \frac{{r}_{i+\tfrac{5}{2}}^{2}{\left(\tfrac{{r}_{i+\tfrac{3}{2}}}{{r}_{i+\tfrac{5}{2}}}\right)}^{{\ell }}-{r}_{i-\tfrac{3}{2}}^{2}}{2-{\ell }},\end{array}\end{eqnarray} \tag{ 186 }$$

if ${\ell }\ne 2$, and

$$\begin{eqnarray}\begin{array}{rcll}{\phi }_{\mathrm{out}i+\displaystyle \frac{1}{2},k+\displaystyle \frac{1}{2}}({\ell },m) & = & {\phi }_{\mathrm{out}i+\displaystyle \frac{3}{2},k+\displaystyle \frac{1}{2}}({\ell },m){\left(\displaystyle \frac{{r}_{i+\tfrac{3}{2}}}{{r}_{i+\tfrac{5}{2}}}\right)}^{{\ell }} & \,+{B}_{r}\left(r^{\prime} ,{\ell },m,{\phi }_{k+\tfrac{1}{2}}\right){r}_{i+\tfrac{3}{2}}^{{\ell }}\mathrm{ln}\left(\displaystyle \frac{{r}_{i+\tfrac{5}{2}}}{{r}_{i+\tfrac{3}{2}}}\right),\end{array}\end{eqnarray} \tag{ 187 }$$

if ${\ell }=2$, and

$$\begin{eqnarray}&&{\phi }_{\mathrm{out}\displaystyle \frac{1}{2},k+\displaystyle \frac{1}{2}}(0.0)={\phi }_{\mathrm{out}\displaystyle \frac{3}{2},k+\displaystyle \frac{1}{2}}(0.0)+\displaystyle \frac{1}{2}\mathrm{Re}A(1,0,0){r}_{i+\tfrac{5}{2}}^{2}\end{eqnarray} \tag{ 188 }$$

if ${\ell }=0$. The gravitational potential is finally calculated as

$$\begin{eqnarray}\displaystyle \begin{array}{rcl}{e}_{\mathrm{grav},i+\displaystyle \frac{1}{2},j+\displaystyle \frac{1}{2},k+\displaystyle \frac{1}{2}} & = & \sum _{{\ell }=0}^{{N}_{{\ell }}}\sum _{m=0}^{{\ell }}\displaystyle \frac{-G}{2{\ell }+1}{\tilde{P}}_{{\ell }}^{m}(\cos {\theta }_{j})\left({\phi }_{\mathrm{in}i-\displaystyle \frac{1}{2}}({\ell },m,k)+{\phi }_{\mathrm{out}i-\displaystyle \frac{1}{2}}({\ell },m,k\right)\left\{\begin{array}{cc}1 & {\ell }=0\\ \sqrt{2} & {\ell }\ne 0.\end{array}\right.\end{array}\end{eqnarray} \tag{ 189 }$$

To assess the accuracy of the gravitational potential expansion as a function of multipole number and to verify the code, the gravitational potentials computed by Equation (189) for a Maclaurin spheroid are compared to those given by an exact analytic solution in Section 5.7.

The ability of a supernova code to simulate a spherical outgoing shock is an important first test. An analytic solution for a spherical outgoing shock is available for the "point-blast explosion" problem, which consists of the instantaneous deposition of an amount of energy, E₀, at a point in a zero-gravity, stationary, uniform medium of constant density, ${\rho }_{0}$. The energy E₀ is required to be very large in comparison with the initial energy of the medium. The analytic solution for this problem was found by Taylor (1950) and Sedov (1959), and various parts of the solution can be found in a number of text books, such as Mihalas & Mihalas (1984) or Zel'dovich & Raizer (1967), with the most complete account being given in Landau & Lifshitz (1959). Because of typos in the publications cited above, we have rederived the solution and have found that Equation (99.8) of Landau & Lifshitz (1959) should be multiplied by the square of the normalized radius (see below) and that the expression for ${\nu }_{5}$ in Equation (99.10) of Landau & Lifshitz (1959) should be replaced by

$$\begin{eqnarray}&&{\nu }_{5}=\displaystyle \frac{2}{\gamma -1}.\end{eqnarray} \tag{ 190 }$$

To set up this problem, a uniform, $\gamma =5/3$ gas maintained in hydrostatic equilibrium within a spherical volume by an external pressure boundary condition and having a constant density ${\rho }_{0}=0.1$ ${{\rm{g\; cm}}}^{-3}$ was divided into 200 zones of equal, 1 cm width. The gas was given a constant ambient temperature of 10⁻⁸ MeV. A point explosion at the center of the spherical mass was simulated by instantaneously depositing an energy, ${E}_{0}\simeq 6.06\times {10}^{17}$ erg, by increasing the temperature of the first zone to 1 MeV. Rather than using a simple gamma-law EoS, we used the Chimera non-NSE EoS, consisting of half neutrons and half protons. The electron and photon contributions were set to zero. In this way, the EoS used was equivalent to a gamma-law EoS with $\gamma =5/3$, but the Chimera EoS machinery (e.g., composition remapping, EoS interpolation) described in Section 3 was tested as well.

Denoting the distance of the shock from the origin by the term r_s, the time-dependence of r_s is given by

$$\begin{eqnarray}&&{r}_{s}={\xi }_{s}{\left(\displaystyle \frac{{E}_{0}{t}^{2}}{{\rho }_{0}}\right)}^{1/5},\end{eqnarray} \tag{ 191 }$$

where ${\xi }_{s}$ is determined by

$$\begin{eqnarray}&&{\xi }_{s}^{5}\displaystyle \frac{32\pi }{25\left({\gamma }^{2}-1\right)}{\displaystyle \int }_{0}^{1}\left({\xi }^{4}\displaystyle \frac{\rho }{{\rho }_{s}}{\left(\displaystyle \frac{u}{{u}_{s}}\right)}^{2}+{\xi }^{9}\displaystyle \frac{p}{{p}_{s}}\right)d\xi =1,\end{eqnarray} \tag{ 192 }$$

where u, ρ, and p are the fluid velocity, density, and pressure, respectively, between the origin and the shock. The subscript "s" denotes their immediate post-shock values. Numerically integrating Equation (192), using Equations (99.10) of Landau & Lifshitz (1959), gives ${\xi }_{s}=1.17$.

A comparison of r_s as a function of time, computed by Chimera versus the analytic result given by Equation (191) is shown in Figure 14(a) and demonstrates agreement to within a few percent. Figure 14(b) shows that the velocity, density, and pressure of the fluid behind the shock normalized by their immediate post-shock values also agree well with the analytic solutions.

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A standard hydrodynamics problem admitting an analytic solution is the Sod shock tube problem (Sod 1978). As Sod's formulation was done in the context of a plane-parallel geometry, we approximate such a geometry with Chimera's spherical grid by working with a variable $x=R-r$, where $R={10}^{5}$ cm and −2 ≤ r ≤ 2 cm. We cover the $-2\leqslant r\leqslant 2$ range of r with a grid consisting of 200 zones, initially equally spaced. Following Sod's original formulation (Sod 1978), we set up a Riemann problem with pressure, p = 1 erg cm⁻³, density, $\rho =1$ ${{\rm{g\; cm}}}^{-3}$, velocity, u = 0 ${{\rm{cm\; s}}}^{-1}$, for $x\lt 0$, and p = 0.1 erg cm⁻³, $\rho =0.125$ ${{\rm{g\; cm}}}^{-3}$, and u = 0 ${{\rm{cm\; s}}}^{-1}$, for $x\gt 0$. The same EoS was used as described above for the point-blast problem, resulting in a constant γ of 5/3. The results of the test at time $t=0.5\,{\rm{s}}$ are shown in Figure 15.

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Numerical hydrodynamics schemes employing Riemann solvers, such as the scheme employed in Chimera, can introduce low-amplitude, post-shock oscillations in flows involving shocks unless extra dissipation is added. To suppress these, Chimera introduces a small amount of dissipation by reducing locally the order of the interpolation scheme in the neighborhood of sufficiently strong shocks (see Colella & Woodward 1984, Section 4 and Appendix). In particular, the left- and right-hand states are modified by Equations (4.1) of Colella & Woodward (1984), namely

$$\begin{eqnarray}\begin{array}{rcl}{a}_{L,i}^{\mathrm{flat}} & = & {a}_{i}^{n}{f}_{i}+{a}_{L,i}(1-{f}_{i}),\\ {a}_{R,i}^{\mathrm{flat}} & = & {a}_{i}^{n}{f}_{i}+{a}_{R,i}(1-{f}_{i}),\end{array}\end{eqnarray} \tag{ 193 }$$

where the "flattening parameter" $(0\leqslant {f}_{i}\leqslant 1)$ determines the mixture of first-order and higher-order PPM interpolations in constructing the left and right states. For the test results shown in Figure 15(a), Chimera was run in Lagrangian mode with the Colella & Woodward (1984) suggested value of 0.75 for ω⁽¹⁾ in the equation following their Equation (A.2), and a maximum value of the flattening parameter, f_i = 0.5. As can be seen, the agreement between the analytical and the numerical results is very good, but very-low-amplitude, post-shock oscillations are evident, particularly in the velocity. In Figure 15(b), we show the results of the same test except that the parameter ω⁽¹⁾ was set to 0.6, and the maximum value of the flattening parameter was set to 1. This introduces more dissipation, and there is now no evidence of any post-shock oscillations in the numerical solutions. Finally, Figure 15(c) shows the results of the test with Chimera now run in Eulerian mode with the Colella & Woodward (1984) suggested value of ω⁽¹⁾ and the maximum value of f_i set to 0.5, as in the test shown in Figure 15(a). The agreement between the numerical and analytical results is again very good, and there is no sign of post-shock oscillations.

A test suggested by Shu & Osher (1989) involves structure, testing the resolution of the numerical hydrodynamics scheme: A moving shock interacts with sine waves in density. Initially, ρ = 3.85713 ${{\rm{g\; cm}}}^{-3}$, v = 2.639369 ${{\rm{cm\; s}}}^{-1}$, and P = 10.33333 erg cm⁻³, for x < −0.8, and ρ = 1 + 0.2sin 5x ${{\rm{g\; cm}}}^{-3}$, v = 0 ${{\rm{cm\; s}}}^{-1}$, and P = 1 erg cm⁻³, for x > −0.8. The results are plotted in Figure 16. The black line shows the solution obtained with a grid of 3200 evenly spaced zones, which is taken as the reference solution. The dashed green line and the red "X"s show the solution obtained with a grid of 800 and 200 zones, respectively. Clearly, the solution has converged with a grid of 800 zones. With 200 zones, the solution still shows the detailed structure, albeit with somewhat reduced amplitude.

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A test of the radial advection and, in particular, the geometry corrections for a spherical geometry, given the choice of such a geometry in Chimera, consists of solving the Euler equations with an adiabatic Γ-law EoS for an initial self-similar radial outflow problem (Mignone 2014) described by

$$\begin{eqnarray}&&\rho (\xi ,0)={\rho }_{0}(\xi ),\quad u(\xi ,0)={\alpha }_{0}\xi ,\quad p(\xi ,0)=\displaystyle \frac{1}{{\rm{\Gamma }}},\end{eqnarray} \tag{ 194 }$$

where ρ(ξ,0) is an arbitrary function and α₀ is a constant. An exact analytic solution of this problem is given, for spherical symmetry, by

$$\begin{eqnarray}\begin{array}{rclllcl}\rho (\xi ,t) & = & {\left(\displaystyle \frac{\alpha (t)}{{\alpha }_{0}}\right)}^{3},\quad u(\xi ,t)={\alpha }_{t}\xi , & \,p & (\xi ,0) & = & \displaystyle \frac{1}{{\rm{\Gamma }}}{\left(\displaystyle \frac{\alpha (t)}{{\alpha }_{0}}\right)}^{3{\rm{\Gamma }}},\quad {\rm{with}}\quad \alpha (t)={\alpha }_{0}/(1+{\alpha }_{0}t).\end{array}\end{eqnarray} \tag{ 195 }$$

A test that focuses on the remapping procedure is that of a self-similar outflow at constant density with the imposition of constant pressure, as described by Blondin & Lundqvist (1993) and Mignone (2014). Following their example, we set α₀ = 100 and utilize a grid of 100 evenly spaced zones. The solution is characterized by constant velocity/radius and constant density. The solution of this problem for the first 10 zones, where the geometry-dependent corrections in the remap are most important, is shown in Figure 17. The numerical solution exhibits a constant velocity/radius that matches the analytic solution.

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As a test of the angular advection algorithms and the logic involved in deflashing a zone (transitioning it from NSE to non-NSE) and flashing a zone (transitioning it from non-NSE to NSE), an electron fraction ${Y}_{{\rm{e}}}$ pattern was advected in angle (θ) across most of 256 angular zones, beginning at zone 18 (θ = 0.22) and continuing across the grid. The ${Y}_{{\rm{e}}}$ pattern consisted of a linear rise over 10 zones, from a value of 0.3472 to a value of 0.4960. A density of $2.47\times {10}^{7}\,{{\rm{g\; cm}}}^{-3}$ and a temperature of 0.45 MeV were chosen, which are representative of conditions where material flows in or out of NSE. Between zones 80 and 81 (θ = 0.98), a transition from NSE to non-NSE was imposed, and between zones 168 and 169 (θ = 2.1), a transition from non-NSE to NSE was imposed. As can be seen in Figure 18(a), the pattern in ${Y}_{{\rm{e}}}$ is nicely preserved as it is advected across the angular grid. A similar test with similar results performed for the azimuthal advection algorithms is shown in Figure 18(b).

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When run in normal mode, the Chimera hydrodynamics does not impose total energy conservation, where the total energy is defined by the integral on the left-hand side of Equation (99); therefore, its ability to conserve total energy is a rigorous test of the hydrodynamics algorithms. As a test of Chimera's ability to conserve energy with the realistic EoS described in Section 3.2 and its numerical implementation, we performed two Newtonian hydrodynamics simulations initiated from a 15 ${M}_{\odot }$ Woosley & Heger (2007) progenitor and carried out for 2 s post-bounce, at which point the bounce shock had traversed 38,000 of our 43,000 km extent of the radial grid.

In the first simulation, referred to as NH_par, Chimera was run in its normal mode. Away from shocks, the specific internal energy was updated during the Lagrangian step by the first law, Equation (124), and remapped as described by Equation (132). In the vicinity of a shock, the specific total energy, defined immediately above Equation (121), was evolved during the Lagrangian step by Equation (121), Equation (134), and Equation (135), and remapped by an equation analogous to Equation (132), with the specific internal energy then extracted by Equation (136). In the second simulation, referred to as NH_tot, the specific total energy was updated during the Lagrangian step and remapped for all zones, whether they were in the vicinity of a shock or not. Run in this mode, Chimera automatically conserves the total energy apart from the source term given by Equation (134), which is added to the total energy after the Lagrangian step. Total energy conservation during the Lagrangian step can be ascertained by multiplying Equation (121) by the mass ${V}_{i}^{n+\tfrac{1}{2}}{\rho }_{i}^{n+\tfrac{1}{2}}$ and noting that the terms involving the pressure cancel in pairs when summed over the zones, leaving only the surface terms, and noting also that the term ${f}_{\nu i}^{n}$ is zero for these hydrodynamics-only runs. During the remap step, the terms on the right-hand side of Equation (132) cancel in pairs after multiplying by the zone mass $\delta {{ \mathcal M }}_{i}^{n+1}$. Careful bookkeeping kept track of all nonphysical changes in the specific energy having no dynamical effect, such as occurs most importantly when material is advected between adjacent spatial zones characterized by different EoSs with slightly different energy zeros, as described in Section 4.4.3.

The results of this test are shown in Figure 19. The Lagrangian trajectories, at 0.025 ${M}_{\odot }$ intervals, for the NH_tot simulation shown in Figure 19(a) were almost identical to those of the NH_par simulation and, thus, are not shown. The shock trajectories (red and dashed green lines) for the two simulations, shown in both panels of Figure 19 are essentially on top of each other. The total energy (blue and violet lines) is shown in Figure 19(b). The same arbitrary constant has been added to the energy of each simulation to bring the energies within the range of the energy ordinate. They are gratifyingly flat, with both showing a small blip at bounce, and the NH_par simulation (blue line) shows a small rise of about 10⁴⁹ erg from 700 ms to the end of the simulation. In that simulation, there is also a small decline in total energy from bounce to about 700 ms, which we traced to the remapping of the velocity, thus conserving momentum by design, rather than remapping the square of the velocity, which would conserve kinetic energy by design. This slight decline in kinetic energy does not appear in the evolved total energy of the NH_tot simulation (violet line). In this case, the specific total energy is the remapped quantity and therefore automatically conserved, modulo the small source term on the right-hand side of the specific total energy equation, Equation (134), as noted above. Figures 20(a) and 20(b) plot the evolution of the gravitational, internal, and kinetic energy components of the total energy for the simulations NH_par (dashed lines) and NH_tot (solid lines). Figure 20(a) plots the evolution of the energy components of the two simulations but at a scale for which differences are not readily discernible. Figure 20(b) plots the last 200 ms of the two simulations at a much finer scale. The same arbitrary constant has been added to each pair of energy components of the two simulations so they fit within the energy range of the ordinate. It is seen that the energy differences are small, with most of the energy differences arising from the ∼3 × 10⁴⁹ erg difference in the internal energy and the ∼1 × 10⁴⁹ erg difference in the kinetic energy.

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Finally, an important quantity relating to the explosion energy obtained in CCSNe simulations and frequently used in comparing simulation outcomes between groups is the "diagnostic energy" E_diag, which is the sum of the gravitational, thermal, and kinetic energy in each zone in turn summed over all zones for which the sum in that zone is positive. At 2500 ms from the initiation of core collapse, the diagnostic energy for both of the above models had essentially become constant and was 0.477 B for the NH_tot simulation and 0.475 B for the NH_par simulation.

Both simulations, NH_par and NH_tot, were performed on an adaptive radial grid of 720 zones, as described in Section 4.5. To ascertain whether this grid resolution results in a radially converged numerical solution, we have performed the NH_par simulation with 240 and 480 zones. The results are shown in Figure 21. It is clear that the numerical solutions have essentially converged at a radial grid resolution of 480 zones but not with a radial grid resolution of 240 zones.

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To verify the accuracy of the gravitational potential expansions given in Sections 4.6.2 and 4.6.3, and to ascertain an appropriate maximum number of multipoles to use in simulations, we have compared with an analytic solution the results of the axisymmetric, Equations (147) and (148), and non-axisymmetric, Equations (187)–(189), expansions for the gravitational potential of a Maclaurin spheroid. We consider the spheroid described by Couch et al. (2013), viz., a spheroid of uniform density of ρ = 1 ${{\rm{g\; cm}}}^{-3}$ embedded in a background of vanishing density. The spheroid has a semimajor axis of 1 m, an eccentricity of 0.9, and is located in a spherical volume of radius 2 m. So that Equations (147), (148), and (187)–(189) can be used to compute the gravitational potential, we set up a spherical computational grid with the spheroid located at the center. A 720 × 240 grid is used for the axisymmetric multipole expansion, reflecting the grid resolution in some of our recent 2D simulations, and a grid of 540 × 180 × 180 is used for the non-axisymmetric multipole expansion, reflecting the angular resolution used in some of our 3D simulations. Where the boundary of the spheroid cuts through a given zone, the density of that zone is adjusted in proportion to the percentage of the zone interior to the boundary. The analytic solution that we use for a point interior to the spheroid is given by Equation (21) of Couch et al. (2013), and for a point exterior to the spheroid, we use the solution given by Equation (1) of Hofmeister et al. (2018).

The results of the comparisons are shown in Figure 22. The zone-weighted mean of the deviation from the analytic solutions of our multipole expansions decreases with ℓ_max, the maximum multipole used, and is below 0.02 percent for ℓ_max ≥ 10. The maximum deviation also decreases with ℓ_max and is below 1% for ℓ_max ≥ 10. The choice of ℓ_max in a simulation is obviously a compromise between accuracy and computational time. Chimera simulations typically use a value of ℓ_max = 10.

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The time evolution of the invariant occupation function $f=f(x,p)$ [number of neutrinos per state at the phase point $(x,p)]$ is given by the coordinate invariant Boltzmann equation (e.g., Lindquist 1966)

$$\begin{eqnarray}&&\displaystyle \frac{{df}}{d{\ell }}=\displaystyle \frac{{{dx}}^{\alpha }}{d{\ell }}\displaystyle \frac{\partial f}{\partial {x}^{\alpha }}+\displaystyle \frac{{{dp}}^{\alpha }}{d{\ell }}\displaystyle \frac{\partial f}{\partial {p}^{\alpha }}={\left(\displaystyle \frac{{df}}{d{\ell }}\right)}_{S},\end{eqnarray} \tag{ 198 }$$

where ℓ is the affine path-length, which we choose to be defined such that

$$\begin{eqnarray}&&{p}^{\alpha }=\displaystyle \frac{{{dx}}^{\alpha }}{d{\ell }},\end{eqnarray} \tag{ 199 }$$

and the right-hand side of Equation (198) denotes the change in f due to sources—i.e., emission, absorption, and scattering. In the spirit of the RbR approximation in which transport along each radial ray is assumed to be spherically symmetric, we will consider the spherically symmetric form of the transport equation. In addition, we will express the Boltzmann equation in the comoving or fluid frame with the intention of executing our neutrino transport along with the Lagrangian radial hydrodynamics step, to be followed by a remap of both the matter and the neutrino quantities to the original grid or to the grid displaced according to a regridding algorithm. We will hereafter denote all quantities defined with respect to the fluid frame by a subscript "0."

The derivation, leading to Equation (215), of the Boltzmann transport equation for spherically symmetric spacetimes, beginning with Equation (198), can be found with varying detail in a number of references (e.g., Lindquist 1966; Castor 1972; Mihalas & Mihalas 1984; Baron et al. 1989; Mezzacappa & Matzner 1989), and we include enough detail so that important quantities used subsequent to Equation (215) are clearly defined. We assume that neutrinos follow null geodesics between localized interactions, so that their paths between interactions are given by

$$\begin{eqnarray}\begin{array}{rcl}0 & = & \displaystyle \frac{{{dp}}_{0}^{\alpha }}{d{\ell }}+\left\{\begin{array}{c}\alpha \\ \beta \ \ \gamma \end{array}\right\}{p}_{0}^{\beta }\displaystyle \frac{{{dx}}^{\gamma }}{d{\ell }}\,\Rightarrow \,\displaystyle \frac{{{dp}}_{0}^{\alpha }}{d{\ell }}=-\left\{\begin{array}{c}\alpha \\ \beta \ \ \gamma \end{array}\right\}{p}_{0}^{\beta }{p}_{0}^{\gamma },\end{array}\end{eqnarray} \tag{ 200 }$$

where $\left\{\begin{array}{c}\alpha \\ \beta \ \ \gamma \end{array}\right\}$ is a Christoffel symbol of the second kind (connection coefficients in the coordinate basis), usually denoted by ${{\rm{\Gamma }}}_{\beta \gamma }^{\alpha }$, but we reserve the latter symbol to denote the Ricci rotation coefficients (connection coefficients in the orthonormal basis).

To evaluate the source functions, we choose a local, comoving, orthonormal set of basis vectors (${{\boldsymbol{e}}}_{t},{{\boldsymbol{e}}}_{m},{{\boldsymbol{e}}}_{\theta },{{\boldsymbol{e}}}_{\phi }$) parallel to a spherical polar coordinate basis to resolve the components of the neutrino four-momentum. In terms of this orthonormal set of basis vectors, the components of four-vectors, such as the four-momentum, ${p}_{0}^{\hat{{\rm{a}}}}$, are denoted by characters with hatted Latin indices. In terms of the original components, ${p}_{0}^{\alpha }$, the components with respect to the orthonormal basis are given by

$$\begin{eqnarray}&&{p}_{0}^{\hat{{\rm{a}}}}={\epsilon }_{\alpha }^{\hat{{\rm{a}}}}{p}_{0}^{\alpha }\,\text{with inverse}\,{p}_{0}^{\alpha }={\epsilon }_{\hat{{\rm{a}}}}^{\alpha }{p}_{0}^{\hat{{\rm{a}}}}={\left({\epsilon }_{\alpha }^{\hat{{\rm{a}}}}\right)}^{-1}{p}_{0}^{\hat{{\rm{a}}}}.\end{eqnarray} \tag{ 201 }$$

Substituting the second part of Equation (201) into Equation (200) and rearranging the indices, ${{dp}}^{\hat{{\rm{a}}}}/d{\ell }$ is given by

$$\begin{eqnarray}\begin{array}{rclcl}\displaystyle \frac{{{dp}}_{0}^{\hat{{\rm{a}}}}}{d{\ell }} & = & -{\epsilon }_{\alpha }^{\hat{{\rm{a}}}}{\epsilon }_{\hat{{\rm{c}}}}^{\gamma }\left[{\left(\displaystyle \frac{\partial ({\epsilon }_{\hat{{\rm{b}}}}^{\alpha })}{\partial {x}^{\gamma }}\right)}_{{x}^{\gamma \ne \alpha }}+\left\{\begin{array}{c}\alpha \\ \beta \ \ \gamma \end{array}\right\}{\epsilon }_{\hat{{\rm{b}}}}^{\beta }\right]{p}_{0}^{\hat{{\rm{b}}}}{p}_{0}^{\hat{{\rm{c}}}} & = & -{{\rm{\Gamma }}}_{\hat{{\rm{b}}}\hat{{\rm{c}}}}^{\hat{{\rm{a}}}}{p}_{0}^{\hat{{\rm{b}}}}{p}_{0}^{\hat{{\rm{c}}}},\end{array}\end{eqnarray} \tag{ 202 }$$

where the Ricci rotation coefficients, ${{\rm{\Gamma }}}_{\hat{{\rm{a}}}\hat{{\rm{c}}}}^{\hat{{\rm{b}}}}$, are defined by Equation (202). Using Equations (199), (201), and (202), Equation (198), written in terms of the ${p}^{\hat{{\rm{a}}}}$, becomes

$$\begin{eqnarray}&&{p}_{0}^{\hat{{\rm{b}}}}\left[{\epsilon }_{\hat{{\rm{b}}}}^{\alpha }{\left(\displaystyle \frac{\partial {f}_{0}}{\partial {x}^{\alpha }}\right)}_{{x}^{\beta \ne \alpha },{p}_{0}^{\hat{{\rm{c}}}}}-{{\rm{\Gamma }}}_{\hat{{\rm{b}}}\hat{{\rm{c}}}}^{\hat{{\rm{a}}}}{p}_{0}^{\hat{{\rm{c}}}}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial {p}_{0}^{\hat{{\rm{a}}}}}\right)}_{{x}^{\gamma },{p}_{0}^{\hat{{\rm{d}}}\ne \hat{{\rm{a}}}}}\right]={\left(\displaystyle \frac{\delta {f}_{0}}{\delta {\ell }}\right)}_{S}.\end{eqnarray} \tag{ 203 }$$

The components of ${p}_{0}^{\hat{{\rm{a}}}}$ with respect to the above orthonormal basis are given by

$$\begin{eqnarray}\begin{array}{rcll}({p}_{0}^{\hat{0}},{p}_{0}^{\hat{1}},{p}_{0}^{\hat{2}},{p}_{0}^{\hat{3}}) & \equiv & ({p}_{0}^{\hat{0}},{\bar{p}}_{0})=\displaystyle \frac{1}{c}\left[{\epsilon }_{0},{\epsilon }_{0}{\mu }_{0},{\epsilon }_{0}\sqrt{1-{\mu }_{0}^{2}}\cos {\phi }_{p},\right. & \left.{\epsilon }_{0}\sqrt{1-{\mu }_{0}^{2}}\sin {\phi }_{p}\right],\end{array}\end{eqnarray} \tag{ 204 }$$

where ${\epsilon }_{0}$ and ${\mu }_{0}$ are the neutrino energy and the direction cosine of the neutrino three-momentum with respect to the radial direction, respectively, both measured in the comoving frame. Because of the mass shell condition

$$\begin{eqnarray}&&{p}_{0}^{\hat{0}}=\sqrt{{\left({p}_{0}^{\hat{1}}\right)}^{2}+{\left({p}_{0}^{\hat{2}}\right)}^{2}+{\left({p}_{0}^{\hat{3}}\right)}^{2}},\end{eqnarray} \tag{ 205 }$$

only three of the four ${p}_{0}^{\hat{a}}$'s are independent. We choose these independent components to be ${p}_{0}^{\hat{{\imath }}},\hat{{\imath }}=1,2,3$. In terms of these independent four-momentum components, the derivatives, $\partial /\partial {p}_{0}^{\hat{{\imath }}}$ are given by

$$\begin{eqnarray}\begin{array}{rcl}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial {p}_{0}^{\hat{1}}}\right)}_{{p}_{0}^{2^{\prime} },{p}_{0}^{3^{\prime} }} & = & c{\mu }_{0}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial {\epsilon }_{0}}\right)}_{{\mu }_{0}}+\displaystyle \frac{c(1-{\mu }_{0}^{2})}{{\epsilon }_{0}}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial {\mu }_{0}}\right)}_{{\epsilon }_{0}},\\ {\left(\displaystyle \frac{\partial {f}_{0}}{\partial {p}_{0}^{\hat{2}}}\right)}_{{p}_{0}^{1^{\prime} },{p}_{0}^{3^{\prime} }} & = & c{\left(1-{\mu }_{0}^{2}\right)}^{1/2}\cos ({\phi }_{p}){\left(\displaystyle \frac{\partial {f}_{0}}{\partial {\epsilon }_{0}}\right)}_{{\mu }_{0}}+c{\left(1-{\mu }_{0}^{2}\right)}^{1/2}\cos ({\phi }_{p}){\left(\displaystyle \frac{\partial {f}_{0}}{\partial {\mu }_{0}}\right)}_{{\epsilon }_{0}},{\rm{and}}\\ {\left(\displaystyle \frac{\partial {f}_{0}}{\partial {p}_{0}^{\hat{3}}}\right)}_{{p}_{0}^{1^{\prime} },{p}_{0}^{2^{\prime} }} & = & c{\left(1-{\mu }_{0}^{2}\right)}^{1/2}\sin ({\phi }_{p}){\left(\displaystyle \frac{\partial {f}_{0}}{\partial {\epsilon }_{0}}\right)}_{{\mu }_{0}}+c{\left(1-{\mu }_{0}^{2}\right)}^{1/2}\sin ({\phi }_{p}){\left(\displaystyle \frac{\partial {f}_{0}}{\partial {\mu }_{0}}\right)}_{{\epsilon }_{0}}.\end{array}\end{eqnarray} \tag{ 206 }$$

We choose a synchronous gauge with a spherically symmetric, orthogonal metric given by

$$\begin{eqnarray}\begin{array}{rcl}{{ds}}^{2} & = & {g}_{\alpha \beta }{{dx}}^{\alpha }{{dx}}^{\beta }\,=\,-\,{a}^{2}(t,m){c}^{2}{{dt}}^{2}+{b}^{2}(t,m){{dm}}^{2}+{R}^{2}(t,m)\left(d{\theta }^{2}+{\sin }^{2}\theta \,d{\phi }^{2}\right),\end{array}\end{eqnarray} \tag{ 207 }$$

where ${x}^{0}={ct}$, ${x}^{1}=m$, ${x}^{2}=\theta$, and ${x}^{3}=\phi$. With this metric, the transformation functions relating coordinate and orthonormal bases are given by

$$\begin{eqnarray}\begin{array}{rcllcl}{\epsilon }_{\hat{0}}^{0} & = & \displaystyle \frac{1}{a},\quad {\epsilon }_{\hat{1}}^{1}=\displaystyle \frac{1}{b},\quad {\epsilon }_{\hat{2}}^{2}=\displaystyle \frac{1}{R},\quad {\epsilon }_{\hat{3}}^{3}=\displaystyle \frac{1}{R\sin \theta },\, & {\epsilon }_{\hat{a}}^{\beta } & = & 0\ \mathrm{if}\ \hat{a}\ne \beta .\end{array}\end{eqnarray} \tag{ 208 }$$

We choose this form of the metric so that the relativistic equations will closely parallel the Newtonian fluid equations, and various levels of Newtonian approximations can be easily made. The metric function $a=a(t,m)$ is the lapse function and relates the interval of proper time of an observer attached to the motion of a fluid element to an interval of coordinate time, and defined so that coordinate and proper time are equal at infinity. The metric function $b=b(t,m)$ will be chosen so that the coordinate m can be identified with the enclosed rest mass. The metric function R is the areal radius (i.e., the two-sphere area $=4\pi {R}^{2}$). The condition that ties the coordinate system to the comoving frame is that the four-velocity of the fluid, ${u}^{\nu }$, be given by ${u}^{\nu }\equiv {{dx}}^{\nu }/d\tau \equiv c\,{{dx}}^{\nu }/{ds}=[{u}^{0},0,0,0]$, which with the metric given by Equation (207) requires that ${u}^{0}=c/a$. This definition of ${u}^{\nu }$ implies that

$$\begin{eqnarray}&&u=\displaystyle \frac{1}{a}\displaystyle \frac{{dR}}{{dt}},\end{eqnarray} \tag{ 209 }$$

where u is the first component of the four-velocity as observed from a frame of constant areal radius R (May & White 1967). To specify b so that m can be identified with the enclosed rest mass, we note that the rest mass density ρ satisfies the local conservation condition

$$\begin{eqnarray}\begin{array}{rclrcl}0 & = & {(\rho {u}^{\alpha })}_{;\alpha }=\displaystyle \frac{1}{\sqrt{g}}\displaystyle \frac{\partial }{\partial {x}^{\nu }}\left(\sqrt{g}\rho {u}^{\nu }\right)\quad \Rightarrow \quad 0=\displaystyle \frac{\partial }{\partial t}\left({{bR}}^{2}\rho \right) & & & \Rightarrow \quad {{bR}}^{2}\rho ={\rm{constant}},\end{array}\end{eqnarray} \tag{ 210 }$$

where the semicolon denotes covariant differentiation and where we have used the expression for ${u}^{\nu }$ given immediately above Equation (209).

With the metric (Equation (207)), the proper volume, dV, is given by ${dV}=4\pi {R}^{2}b\,{dm}$, or, in terms of the rest mass dm contained in dV, by ${dV}={dm}/\rho$. It follows from these two expressions for dV that the requisite choice of b is

$$\begin{eqnarray}&&b=\displaystyle \frac{1}{4\pi {R}^{2}\rho }.\end{eqnarray} \tag{ 211 }$$

With the above choices for a spherical, comoving coordinate system and a comoving, orthonormal four-vector basis, the coordinate invariant volume elements become

$$\begin{eqnarray}&&{dV}=\sqrt{-g}\,{\epsilon }_{\delta \alpha \beta \gamma }{u}^{\delta }{d}_{1}{x}^{\alpha }{d}_{2}{x}^{\beta }{d}_{3}{x}^{\gamma }\to b\,{dm}\,{R}^{2}d{\rm{\Omega }},\end{eqnarray} \tag{ 212 }$$

$$\begin{eqnarray}&&{dP}=\sqrt{-g}\,{\epsilon }_{{ijk}}{u}^{\delta }\displaystyle \frac{{d}_{1}{p}_{0}^{i}\,{d}_{2}{p}_{0}^{j}\,{d}_{3}{p}_{0}^{k}}{{p}_{0}}\to \displaystyle \frac{1}{{c}^{2}}{\epsilon }_{0}\,d{\epsilon }_{0}\,d{{\rm{\Omega }}}_{p},\end{eqnarray} \tag{ 213 }$$

where g is the determinant of the metric, and ${\epsilon }_{\delta \alpha \beta \gamma }$ and ${\epsilon }_{{ijk}}$ are the Levi-Civita alternating symbols. The invariant distribution function, ${f}_{0}$, introduced at the beginning of this section, is defined so that the number of world lines crossing the volume element, dV, with four-momenta in the range dP about ${{\boldsymbol{p}}}_{0}$ is given by (Lindquist 1966)

$$\begin{eqnarray}\begin{array}{rcl}{dN} & = & \displaystyle \frac{1}{{h}^{3}}{f}_{0}(x,{{\boldsymbol{p}}}_{0})(-{{\boldsymbol{p}}}_{0}\cdot {\boldsymbol{u}}){dVdP}\\ & \to & \displaystyle \frac{1}{{\left({hc}\right)}^{3}}{f}_{0}b\,{{dm}{R}}^{2}d{\rm{\Omega }}\,{\epsilon }_{0}^{2}\,d{\epsilon }_{0}\,d{{\rm{\Omega }}}_{p},\end{array}\end{eqnarray} \tag{ 214 }$$

where the right-hand sides of Equations (212)–(214) are the expressions for dV, dP, and dN in our choice of coordinate system and four-vector basis. Finally, evaluating the transformation coefficients from the metric (Equation (207)) and using them for the Ricci coefficients in Equation (203) and using Equations (204) and (206), the Boltzmann equation becomes

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\epsilon }_{0}}{c}\left\{\displaystyle \frac{1}{{ac}}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial t}\right)}_{m,{\epsilon }_{0},{\mu }_{0}}+\displaystyle \frac{{\mu }_{0}}{b}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial m}\right)}_{t,{\epsilon }_{0},{\mu }_{0}}\right.\\ \qquad -\,\left[\displaystyle \frac{1}{b}\displaystyle \frac{\partial \mathrm{ln}a}{\partial m}{\mu }_{0}+\displaystyle \frac{1}{{ac}}\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}{\mu }_{0}^{2}\right.\left.\,+\,\displaystyle \frac{1}{{ac}}\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}(1-{\mu }_{0}^{2})\right]{\epsilon }_{0}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial {\epsilon }_{0}}\right)}_{t,m,{\mu }_{0}}\\ \qquad +\,\left[\displaystyle \frac{{\mu }_{0}}{{ac}}\left(\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}-\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}\right)-\displaystyle \frac{1}{b}\left(\displaystyle \frac{\partial \mathrm{ln}a}{\partial m}-\displaystyle \frac{\partial \mathrm{ln}R}{\partial m}\right)\right]\left.\,(1-{\mu }_{0}^{2}){\left(\displaystyle \frac{\partial {f}_{0}}{\partial {\mu }_{0}}\right)}_{t,m,{\epsilon }_{0}}\right\}\\ \quad =\,{\left(\displaystyle \frac{{{df}}_{0}}{d{\ell }}\right)}_{S}.\end{array}\end{eqnarray} \tag{ 215 }$$

Note that with the substitutions $a={e}^{{\rm{\Phi }}}$, $b={e}^{{\rm{\Lambda }}}$, and ${\rm{\Gamma }}={(1/b)(\partial R/\partial m)}_{t}$, Equation (215) reduces to Equation (3.7) of Lindquist (1966). For economy of notation and for clarity of presentation, we have suppressed the dependency of ${f}_{0}$ on t, m, ${\epsilon }_{0}$, and ${\mu }_{0}$, and the dependencies of the metric variables a, b, Γ, and R on t and m, and will do so with new dependent variables as they are introduced as long as this brings no ambiguity.

We now modify Equation (215) by transforming from independent variables $(t,m,{\epsilon }_{0},{\mu }_{0})$ to $(t,m,{E}_{0},{\mu }_{0})$ where

$$\begin{eqnarray}&&{E}_{0}=a{\epsilon }_{0}.\end{eqnarray} \tag{ 216 }$$

Therefore, with this variable transformation

$$\begin{eqnarray}&&{f}_{0}(t,m,{\epsilon }_{0},{\mu }_{0})={f}_{0}\left(t,m,\displaystyle \frac{{E}_{0}}{a},{\mu }_{0}\right)={f}_{0}^{{\prime} }(t,m,{E}_{0},{\mu }_{0}).\end{eqnarray} \tag{ 217 }$$

Though mathematically imprecise, we will use the same symbol to describe the neutrino distribution functions, originally functions of $(t,m,{\epsilon }_{0},{\mu }_{0})$, as functions of $(t,m,{E}_{0},{\mu }_{0})$. We also define Γ by ${\rm{\Gamma }}={(1/b)(\partial R/\partial m)}_{t}$ so that derivatives with respect to m at constant time can be replaced by derivatives with respect to R, which is directly related to our grid, by the identity

$$\begin{eqnarray}&&{\left(\displaystyle \frac{\partial X}{\partial m}\right)}_{t}={\left(\displaystyle \frac{\partial X}{\partial R}\right)}_{t}{\left(\displaystyle \frac{\partial R}{\partial m}\right)}_{t}=b{\rm{\Gamma }}{\left(\displaystyle \frac{\partial X}{\partial R}\right)}_{t}\equiv b{\rm{\Gamma }}{X}_{,R}.\end{eqnarray} \tag{ 218 }$$

Applying the above definition of Γ and variable transformation to Equation (215) gives

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{E}_{0}}{{ac}}\left\{\displaystyle \frac{1}{{ac}}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial t}\right)}_{m,{\mu }_{0},{E}_{0}}+{\mu }_{0}{\rm{\Gamma }}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial R}\right)}_{t,{\mu }_{0},{E}_{0}}\right.\\ \qquad +\,\displaystyle \frac{1}{{ac}}\left[\displaystyle \frac{\partial \mathrm{ln}a}{\partial t}-{\mu }_{0}^{2}\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}\right.\left.-\,(1-{\mu }_{0}^{2})\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}\right]{E}_{0}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial {E}_{0}}\right)}_{t,m,{\mu }_{0}}\\ \qquad +\,\left[\displaystyle \frac{{\mu }_{0}}{{ac}}\left(\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}-\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}\right)-{\rm{\Gamma }}\left(\displaystyle \frac{\partial \mathrm{ln}a}{\partial R}-\displaystyle \frac{1}{R}\right)\right]\left.\,(1-{\mu }_{0}^{2}){\left(\displaystyle \frac{\partial {f}_{0}}{\partial {\mu }_{0}}\right)}_{t,m,{E}_{0}}\right\}\\ \quad =\,{\left(\displaystyle \frac{{{df}}_{0}}{d{\ell }}\right)}_{{\rm{S}}}.\end{array}\end{eqnarray} \tag{ 219 }$$

The transformation given by Equation (216) affords several advantages. First, with constant values of ${E}_{0k+\tfrac{1}{2}}$ anchoring the energy grid, the energy grid ${\epsilon }_{0k+\tfrac{1}{2}}$ will be scaled to higher values as the lapse dips below unity at high densities—i.e., for constant ${E}_{0k+\tfrac{1}{2}}$, the neutrino energy will scale as ${\epsilon }_{0k+\tfrac{1}{2}}\propto 1/a$. This will permit a smaller upper bound to the neutrino energy grid farther out from the center where $a\simeq 1$, resulting in the grid energies being more closely spaced there for a given number of grid points, while still permitting sufficient energy headroom at high densities to accommodate the high-energy neutrinos that are produced at the high-matter densities prevailing near the core center. Second, the radial derivative of a in the factor multiplying the energy derivative of ${f}_{0}$ in Equation (215) has been replaced by a time derivative in Equation (219). This factor now contains terms involving only time derivatives and, therefore, vanishes for a static spacetime. Thus, apart from energy-changing interactions, for a static spacetime, there will be no flow of neutrinos through the neutrino energy grid as they propagate outward. The gravitational redshifting will consequently be accomplished automatically. In non-static spacetimes, the advection of neutrinos through the energy grid can be performed algebraically (Sections 6.13 and 6.14), another advantage of this scheme. Using this choice of energy gridding, we will assume that the spacetime is constant over a time step and make a small correction to the neutrino distribution at the end of the time step to correct for the change in a during a time step and other processes that shift the neutrinos in energy, such as their advection across spatial zones with differing lapses.

Let the nth angular moment of ${f}_{0}$ be denoted by ${\psi }_{0}^{(n)}$, that is

$$\begin{eqnarray}\begin{array}{rclrcl}{\psi }_{0}^{(n)}(R,t,{E}_{0}) & = & \displaystyle \frac{1}{4\pi }{\displaystyle \int }_{{\rm{\Omega }}}d{\rm{\Omega }}{\mu }_{0}^{n}{f}_{0}(R,t,{E}_{0},{\mu }_{0}) & & = & \displaystyle \frac{1}{2}{\displaystyle \int }_{-1}^{1}d{\mu }_{0}{\mu }_{0}^{n}{f}_{0}(R,t,{E}_{0},{\mu }_{0}),\end{array}\end{eqnarray} \tag{ 220 }$$

and let

$$\begin{eqnarray}\begin{array}{rclrcl}{\left(\displaystyle \frac{{{df}}_{0}}{d{\ell }}\right)}_{S}^{(n)} & \equiv & {\left(\displaystyle \frac{d}{d{\ell }}{f}_{0}(m,t,{E}_{0},{\mu }_{0}\right)}_{S}^{(n)} & & \equiv & \displaystyle \frac{1}{2}{\displaystyle \int }_{-1}^{1}{\mu }_{0}^{n}d{\mu }_{0}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0}(m,t,{E}_{0},{\mu }_{0}\right)}_{S}.\end{array}\end{eqnarray} \tag{ 221 }$$

Then the first two angular moments of Equation (219) are given by

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{E}_{0}}{{ac}}\left\{\displaystyle \frac{1}{{ac}}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial t}\right)}_{m,{E}_{0}}+{\rm{\Gamma }}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(1)}}{\partial R}\right)}_{t,{E}_{0}}\right.\\ \qquad +\,\displaystyle \frac{1}{{ac}}{E}_{0}\left[\displaystyle \frac{\partial \mathrm{ln}a}{\partial t}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial {E}_{0}}\right)}_{t,m}\right.-\,\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(2)}}{\partial {E}_{0}}\right)}_{t,m}-\,\left.\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}{\left(\displaystyle \frac{\partial }{\partial {E}_{0}}\right)}_{t,m}\left({\psi }_{0}^{(0)}-{\psi }_{0}^{(2)}\right)\right]\\ \qquad -\,\displaystyle \frac{1}{{ac}}\left(\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}-\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}\right)\left({\psi }_{0}^{(0)}-3{\psi }_{0}^{(2)}\right)\left.\,+\,{\rm{\Gamma }}\left(-\displaystyle \frac{\partial \mathrm{ln}a}{\partial R}+\displaystyle \frac{1}{R}\right)2{\psi }_{0}^{(1)}\right\}\\ \quad =\,{\left(\displaystyle \frac{{{df}}_{0}}{d{\ell }}\right)}_{S}^{(0)}\end{array}\end{eqnarray} \tag{ 222 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{E}_{0}}{{ac}}\left\{\displaystyle \frac{1}{{ac}}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(1)}}{\partial t}\right)}_{m,{E}_{0}}+{\rm{\Gamma }}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(2)}}{\partial R}\right)}_{t,{E}_{0}}\right.\\\qquad +\,\displaystyle \frac{1}{{ac}}{E}_{0}\left[\displaystyle \frac{\partial \mathrm{ln}a}{\partial t}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(1)}}{\partial {E}_{0}}\right)}_{t,m}\,-\,\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(3)}}{\partial {E}_{0}}\right)}_{t,m}-\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}{\left(\displaystyle \frac{\partial }{\partial {E}_{0}}\right)}_{t,m}\,\left({\psi }_{0}^{(1)}-{\psi }_{0}^{(3)}\right)\right]\\\qquad -\,\displaystyle \frac{1}{{ac}}\left(\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}-\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}\right)\left(2{\psi }_{0}^{(1)}-4{\psi }_{0}^{(3)}\right)\left.\,-\,{\rm{\Gamma }}\left(-\displaystyle \frac{\partial \mathrm{ln}a}{\partial R}+\displaystyle \frac{1}{R}\right)({\psi }_{0}^{(0)}-3{\psi }_{0}^{(2)})\right\}\\\quad =\,{\left(\displaystyle \frac{{{df}}_{0}}{d{\ell }}\right)}_{S}^{(1)}.\end{array}\end{eqnarray} \tag{ 223 }$$

To close this system of moment equations, we employ flux limiting to derive a relation between ${\psi }_{0}^{(1)}$ and ${\psi }_{0}^{(0)}$. In analogy with the procedure described in Levermore & Pomraning (1981), let the scalar Eddington factors ${\eta }^{(n)}$ be defined as the ratio of ${\psi }_{0}^{(n)}$ to ${\psi }_{0}^{(0)}$ so that ${\psi }_{0}^{(n)}={\eta }^{(n)}{\psi }_{0}^{(0)}$. Substituting ${\psi }_{0}^{(n)}={\eta }^{(n)}{\psi }_{0}^{(0)}$ in Equations (222) and (223), solving Equation (222) for ${(\partial {\psi }_{0}^{(0)}/\partial t)}_{m,{E}_{0}}$ and substituting the result into Equation (223) gives

$$\begin{eqnarray}&&\begin{array}{l}\left\{\displaystyle \frac{1}{{ac}}{\left(\displaystyle \frac{\partial {\eta }^{(1)}}{\partial t}\right)}_{m,{E}_{0}}-{\rm{\Gamma }}{\eta }^{(1)}{\left(\displaystyle \frac{\partial {\eta }^{(1)}}{\partial R}\right)}_{t,{E}_{0}}\right.\,+\,\displaystyle \frac{1}{{ac}}\left(\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}-\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}\right){E}_{0}{\left(\displaystyle \frac{\partial {\eta }^{(2)}}{\partial {E}_{0}}\right)}_{t,m}{\eta }^{(1)}\\ \ \ +\,\displaystyle \frac{1}{{ac}}\left(\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}-\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}\right)\left(-{\eta }^{(1)}-3{\eta }^{(1)}{\eta }^{(2)}+4{\eta }^{(3)}\right)\,+\,{\rm{\Gamma }}\left(\displaystyle \frac{\partial \mathrm{ln}a}{\partial R}-\displaystyle \frac{1}{R}\right)\left(1+2{\left({\eta }^{(1)}\right)}^{2}-3{\eta }^{(2)}\right)\\ \ \ +\,{\rm{\Gamma }}{\left(\displaystyle \frac{\partial {\eta }^{(2)}}{\partial R}\right)}_{t,{E}_{0}}+\displaystyle \frac{1}{{ac}}\left(\displaystyle \frac{\partial \mathrm{ln}a}{\partial t}-\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}\right){E}_{0}{\left(\displaystyle \frac{\partial {\eta }^{(1)}}{\partial {E}_{0}}\right)}_{t,m}\,\left.-\,\displaystyle \frac{1}{{ac}}\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}{E}_{0}{\left(\displaystyle \frac{\partial \left({\eta }^{(1)}-{\eta }^{(3)}\right)}{\partial {E}_{0}}\right)}_{t,m}\right\}{\psi }_{0}^{(0)}\\ \ \ +\,\left\{\displaystyle \frac{1}{{ac}}\left(\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}-\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}\right)\left({\eta }^{(1)}{\eta }^{(2)}-{\eta }^{(3)}\right)\right\}{E}_{0}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial {E}_{0}}\right)}_{t,m}\\ \ \ -\,{\rm{\Gamma }}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial R}\right)}_{t,{E}_{0}}{\left({\eta }^{(1)}\right)}^{2}+\displaystyle \frac{{ac}}{{E}_{0}}{\eta }^{(1)}{\left(\displaystyle \frac{{{df}}_{0}}{d{\ell }}\right)}_{S}^{(0)}+{\rm{\Gamma }}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial R}\right)}_{t,{E}_{0}}\,{\eta }^{(2)}=\displaystyle \frac{{ac}}{{E}_{0}}{\left(\displaystyle \frac{{{df}}_{0}}{d{\ell }}\right)}_{S}^{(1)}.\end{array}\end{eqnarray} \tag{ 224 }$$

The factors multiplying ${\psi }_{0}^{(0)}$ and ${(\partial {\psi }_{0}^{(0)}/\partial {E}_{0})}_{t,m}$ in Equation (224) in both the diffusion limit (${\eta }^{(1)}\to 0;$ ${\eta }^{(2)}\to {}^{1}{/}_{3};$ ${\eta }^{(3)}\to 0$) and the free streaming limit (${\eta }^{(n)}\to 1$) are zero. We therefore make the approximation that these two factors are zero everywhere. Equation (224) then becomes

$$\begin{eqnarray}&&\displaystyle \frac{{ac}}{{E}_{0}}{\eta }^{(1)}{\left(\displaystyle \frac{{{df}}_{0}}{d{\ell }}\right)}_{S}^{(0)}+{\rm{\Gamma }}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial R}\right)}_{t,{E}_{0}}\left({\eta }^{(2)}-{\left({\eta }^{(1)}\right)}^{2}\right)=\displaystyle \frac{{ac}}{{E}_{0}}{\left(\displaystyle \frac{{{df}}_{0}}{d{\ell }}\right)}_{S}^{(1)}.\end{eqnarray} \tag{ 225 }$$

Multiplying the right-hand side of Equation (225) by ${\psi }_{0}^{(1)}$/${\psi }_{0}^{(1)}$, i.e., unity, and solving for ${\psi }_{0}^{(1)}$, we get

$$\begin{eqnarray}&&{\psi }_{0}^{(1)}=\displaystyle \frac{-{\rm{\Gamma }}\left({\eta }^{(2)}-{\left({\eta }^{(1)}\right)}^{2}\right){\left(\tfrac{\partial {\psi }_{0}^{(0)}}{\partial R}\right)}_{t,{E}_{0}}-{\eta }^{(1)}\tfrac{{ac}}{{E}_{0}}{\left(\tfrac{{{df}}_{0}}{d{\ell }}\right)}_{S}^{(0)}}{-\tfrac{{ac}}{{E}_{0}}\tfrac{1}{{\psi }_{0}^{(1)}}{\left(\tfrac{{{df}}_{0}}{d{\ell }}\right)}_{S}^{(1)}}.\end{eqnarray} \tag{ 226 }$$

In the diffusion limit, ${\eta }^{(1)}\to 0$ and ${\eta }^{(2)}\to {}^{1}{/}_{3}$, and Equation (226) reduces to the standard diffusion equation if we neglect the second term in the numerator and identify the transport mean free path, ${\lambda }^{(t)}$, as

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\lambda }^{(t)}}=-\displaystyle \frac{{ac}}{{E}_{0}}\displaystyle \frac{1}{{\psi }_{0}^{(1)}}{\left(\displaystyle \frac{{{df}}_{0}}{d{\ell }}\right)}_{S}^{(1)}.\end{eqnarray} \tag{ 227 }$$

In regimes other than the diffusion regime, we regard $\left[{\eta }^{(2)}-{\left({\eta }^{(1)}\right)}^{2}\right]$ as a free parameter, which we write as

$$\begin{eqnarray}&&{ \mathcal F }=3\left[{\eta }^{(2)}-{\left({\eta }^{(1)}\right)}^{2}\right].\end{eqnarray} \tag{ 228 }$$

Using Equations (227) and (228) in Equation (226), we get a diffusion-like equation for ${\psi }_{0}^{(1)}$:

$$\begin{eqnarray}&&{\psi }_{0}^{(1)}=-\displaystyle \frac{{\lambda }^{(t)}}{3}{ \mathcal F }\,{\rm{\Gamma }}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial R}\right)}_{t,{E}_{0}}.\end{eqnarray} \tag{ 229 }$$

Equations (222) and (229) for each energy zone and for each neutrino species (${\nu }_{e}$, ${\bar{\nu }}_{e}$, ${\nu }_{\mu \tau }$, ${\bar{\nu }}_{\mu \tau }$) with a prescription for ${ \mathcal F }$ are the MGFLD equations that are solved in Chimera for the B-series simulations. The modification of this scheme in the vicinity of shocks, used for our C-series and later simulations, is described in Section 6.12. The parameter ${ \mathcal F }$ is referred to as the "flux-limiter," and should be unity in the diffusion limit and tend to zero in such a way that ${\psi }_{0}^{(1)}$  = ${\psi }_{0}^{(0)}$ in the limit of free streaming.

The derivation of Equations (229) and (227) differ from that of the corresponding equations (Equations (A25) and (A26)) in Bruenn (1985). To derive a diffusion-like equation, there the derivative $\partial {\psi }_{0}^{(1)}/\partial t$ in the nonrelativistic version of Equation (223) and all velocity dependent terms were set to zero. However, the results are similar. Apart from the relativistic time dilation factor a, the expressions for $1/{\lambda }_{i}^{(t)}$, given here by Equation (227) and given in Bruenn (1985) by Equation (A26), are the same. Equation (226) for ${\psi }_{0}^{(1)}$ here is similar to Equation (A25) ${\psi }_{0}^{(1)}$ in Bruenn (1985), with the exception here of the relativistic proper distance correction Γ and the factor of $\left[{\eta }^{(2)}-{\left({\eta }^{(1)}\right)}^{2}\right]$, which we take as our flux limiter. The flux limiter in Bruenn (1985) is introduced as a modification of ${\lambda }_{i}^{(t)}$. Finally, the terms in the numerator of Equations (226) here and in Equation (A25) of Bruenn (1985) are neglected, other than the term involving $\partial {\psi }_{0}^{(0)}/\partial R$. They involve the redistribution of neutrinos in angle due to anisotropies in the source terms but are small given that they depend on the product of ${\psi }_{0}^{(0)}$ and ${\psi }_{0}^{(1)}$. Equation (229) is given in differenced form by Equation (274).

The flux limiter, ${ \mathcal F }$, constructed to satisfy the diffusion and free-streaming limits, consists of two parts. The first part is a specific implementation of the usual scheme for interpolating between these two limits, namely

$$\begin{eqnarray}\begin{array}{rcll}{{ \mathcal F }}_{\mathrm{intrp}} & \equiv & {{ \mathcal F }}_{\mathrm{intrp}}(R,t,{\epsilon }_{0})\,= & {\left(1+\displaystyle \frac{1}{3}{\lambda }_{i+\tfrac{1}{2}}^{(t)}({\epsilon }_{0})\left[\displaystyle \frac{\left|{\rm{\Gamma }}\tfrac{\partial {\psi }_{0}^{(0)}({E}_{0})}{\partial R}\right|}{{\psi }_{0}^{(0)}({E}_{0})}\right]\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 230 }$$

From Equations (229) and (230), we see that in the optically thick, diffusion regime, for which ${\lambda }^{(t)}({\epsilon }_{0})\to 0$, ${{ \mathcal F }}_{\mathrm{intrp}}\to 1$ and Equation (229) limits to the diffusion equation

$$\begin{eqnarray}&&{\psi }_{0}^{(1)}({E}_{0})\to -\displaystyle \frac{{\lambda }_{i+\tfrac{1}{2}}^{(t)}({\epsilon }_{0})}{3}{\rm{\Gamma }}\displaystyle \frac{\partial {\psi }_{0}^{(0)}({E}_{0})}{\partial R},\end{eqnarray} \tag{ 231 }$$

while in the optically thin, free-streaming regime, where ${\lambda }^{(t)}({\epsilon }_{0})\to \infty$, Equation (229) limits to the free-streaming condition

$$\begin{eqnarray}&&{\psi }_{0}^{(1)}({E}_{0})={\psi }_{0}^{(0)}({E}_{0}).\end{eqnarray} \tag{ 232 }$$

As it stands, this scheme suffers from the generic problem of an overly rapid transition to the free-streaming limit (i.e., the angular distribution becomes too forwardly peaked) when matter goes from optically thick to optically thin abruptly, such as when the "density cliff" forms in the post-bounce core of a supernova progenitor. To avoid this problem, a second piece of the flux limiter is constructed. It prevents the neutrino angular distribution from becoming more forwardly peaked than the geometrical limit. This geometrical limit can be expressed for $R\gt {R}_{\nu }$, where ${R}_{\nu }\equiv {R}_{\nu }({\epsilon }_{0})$ is the radius of the neutrinosphere for a particular flavor and energy, by ${\psi }_{0}^{(1)}=\tfrac{1}{2}[1+{\mu }_{0\nu }({\epsilon }_{0})]{\psi }_{0}^{(0)}$. This expression assumes that the neutrino distribution function ${f}_{0}$ is constant for rays satisfying ${\mu }_{0}\lt {\mu }_{0\nu }$, and zero otherwise, where ${\mu }_{0\nu }$ is the cosine of the angle subtended between a line extending from a point at R to the neutrinosphere limb and a line extending from the point at R to the core center, as observed by a comoving observer. This geometrical piece of the flux-limiter is then given by

$$\begin{eqnarray}&&{{ \mathcal F }}_{\mathrm{geom}}\equiv {{ \mathcal F }}_{\mathrm{geom}}(R,t,{\epsilon }_{0})=\displaystyle \frac{\tfrac{1}{2}(1+{\mu }_{0\nu }){\psi }_{0}^{(0)}}{\tfrac{1}{3}{\lambda }_{i}^{(t)}\left|{\rm{\Gamma }}\tfrac{\partial {\psi }_{0}^{(0)}}{\partial R}\right|}\,{\rm{if}}\,R\gt {R}_{\nu },\end{eqnarray} \tag{ 233 }$$

and ${{ \mathcal F }}_{\mathrm{geom}}(R,t,{\epsilon }_{0})=1$ interior to the neutrinosphere, $R\leqslant {R}_{\nu }({\epsilon }_{0})$. The comoving angle ${\mu }_{0\nu }$ is given in terms of the fixed frame angle ${\mu }_{\nu }$ by

$$\begin{eqnarray}&&{\mu }_{0\nu }({\epsilon }_{0})=\displaystyle \frac{{\mu }_{\nu }+\beta }{1-{\mu }_{\nu }\beta },\end{eqnarray} \tag{ 234 }$$

where $\beta =v/c$, and the fixed-frame angle ${\mu }_{\nu }$ is given by

$$\begin{eqnarray}&&{\mu }_{\nu }=\sqrt{1-{\left(\displaystyle \frac{{R}_{\nu }}{R}\right)}^{2}{ \mathcal G }},\end{eqnarray} \tag{ 235 }$$

where ${ \mathcal G }$ is the correction for gravitational ray bending given by

$$\begin{eqnarray}&&{ \mathcal G }=\sqrt{\displaystyle \frac{1-2{{GM}}_{g}/{{Rc}}^{2}}{1-2{{GM}}_{g}/{R}_{\nu }{c}^{2}}},\end{eqnarray} \tag{ 236 }$$

where M_g is the gravitational mass. The geometrical flux limiter, ${{ \mathcal F }}_{\mathrm{geom}}$, by itself would relate ${\psi }_{0}^{(1)}$ to ${\psi }_{0}^{(0)}$ outside the neutrinosphere by

$$\begin{eqnarray}&&{\psi }_{0}^{(1)}=\displaystyle \frac{1}{2}\left(1+{\mu }_{0\nu }\right){\psi }_{0}^{(0)}.\end{eqnarray} \tag{ 237 }$$

The final flux limiter is given by

$$\begin{eqnarray}&&{ \mathcal F }=\min [{{ \mathcal F }}_{\mathrm{intrp}},{{ \mathcal F }}_{\mathrm{geom}}].\end{eqnarray} \tag{ 238 }$$

To solve Equation (222) along with Equation (229), we operator split Equation (222) into a transport equation and an energy advection equation

$$\begin{eqnarray}&&{\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial t}\right)}_{m,{E}_{0}}={\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial t}\right)}_{m,{E}_{0}}^{T}+{\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial t}\right)}_{m,{E}_{0}}^{E},\end{eqnarray} \tag{ 239 }$$

where $(\partial {\psi }_{0}^{(0)}/\partial t{)}_{m,{E}_{0}}^{T}$, the rate of change of ${\psi }_{0}^{(0)}$ due to sources and transport, is given by

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{{ac}}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial t}\right)}_{m,{E}_{0}}^{T}+{\rm{\Gamma }}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(1)}}{\partial R}\right)}_{t,{E}_{0}}-{\rm{\Gamma }}\left(\displaystyle \frac{\partial \mathrm{ln}a}{\partial R}-\displaystyle \frac{1}{R}\right)2{\psi }_{0}^{(1)}\,=\,\displaystyle \frac{{ac}}{{E}_{0}}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0}(x,{p}_{0}\right)}_{S}^{(0)},\end{array}\end{eqnarray} \tag{ 240 }$$

and $(\partial {\psi }_{0}^{(0)}/\partial t{)}_{m,{E}_{0}}^{E}$, the rate of change of ${\psi }_{0}^{(0)}$ due to energy advection, is given such that

$$\begin{eqnarray}&&\begin{array}{l}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial t}\right)}_{m,{E}_{0}}^{E}+{E}_{0}\left[\displaystyle \frac{\partial \mathrm{ln}a}{\partial t}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(0)}}{\partial {E}_{0}}\right)}_{t,m}-\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}{\left(\displaystyle \frac{\partial {\psi }_{0}^{(2)}}{\partial {E}_{0}}\right)}_{t,m}\right.\left.\,-\,\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}{\left(\displaystyle \frac{\partial }{\partial {E}_{0}}\right)}_{t,m}\left({\psi }_{0}^{(0)}-{\psi }_{0}^{(2)}\right)\right]\\ \qquad -\,\left[\displaystyle \frac{\partial \mathrm{ln}R}{\partial t}-\displaystyle \frac{\partial \mathrm{ln}b}{\partial t}\right]\left({\psi }_{0}^{(0)}-3{\psi }_{0}^{(2)}\right)=0.\end{array}\end{eqnarray} \tag{ 241 }$$

In advancing ${\psi }_{0}^{(0)}$ and ${\psi }_{0}^{(1)}$ over a time step, the pair of Equations (240) and (229) are solved in one step along with the associated energy and lepton conservation equations introduced below, then Equation (241) is solved in a second step. We will refer to these two separate steps as a transport step and an energy advection step, respectively, and describe in more detail below how each is performed.

In order to derive the Einstein equations, which are needed to obtain expressions for the gravitational mass, M_g, and the metric parameters Γ and a, we need the stress-energy tensor, ${ \mathcal T }\,=$ ${}^{(m)}{ \mathcal T }\,+$ ${}^{(\nu )}{ \mathcal T }$, where ${}^{(m)}{ \mathcal T }$ and ${}^{(\nu )}{ \mathcal T }$ are the matter and neutrino contributions, respectively. Following Mihalas & Mihalas (1984), we begin with the definition of the radiation (i.e., neutrino) stress-energy tensor

$$\begin{eqnarray}&&{}^{(\nu )}{{ \mathcal T }}^{\alpha \beta }=\displaystyle \frac{{c}^{2}}{{h}^{3}}\int \sum _{q}{f}_{0q}\,{p}^{\alpha }{p}^{\beta }\displaystyle \frac{{d}^{3}p}{{{cp}}^{0}},\end{eqnarray} \tag{ 242 }$$

where the sum q is over all neutrino species. In the local comoving orthonormal frame, the components of the stress-energy tensor are

$$\begin{eqnarray}&&{}^{(\nu )}{{ \mathcal T }}^{\hat{{\rm{a}}}\hat{{\rm{b}}}}=\displaystyle \frac{{c}^{2}}{{\left({hc}\right)}^{3}}\int \sum _{q}{f}_{0q}\,{p}_{0}^{\hat{{\rm{a}}}}{p}_{0}^{\hat{{\rm{b}}}}{\epsilon }_{0}d{\epsilon }_{0}\,d{{\rm{\Omega }}}_{0}\end{eqnarray} \tag{ 243 }$$

where, as before, we use Latin indices with hats here to distinguish components with respect to the local comoving orthonormal frame from those with respect to the coordinate basis (distinguished using Greek letters).

Using Equations (204) for ${p}^{\hat{{\rm{a}}}}$ and ${p}^{\hat{{\rm{b}}}}$ in Equation (243) for ${}^{(\nu )}{{ \mathcal T }}^{\hat{{\rm{a}}}\hat{{\rm{b}}}}$, the radiation stress-energy tensor ${}^{(\nu )}{{ \mathcal T }}^{\hat{{\rm{a}}}\hat{{\rm{b}}}}$ can be written in terms of the local neutrino energy density, ${E}_{\nu }$, flux ${F}_{\nu }$, and pressure, ${P}_{\nu }$, as

$$\begin{eqnarray}\begin{array}{rclrcl}\left({E}_{\nu },{F}_{\nu }/c,{P}_{\nu }\right) & = & \displaystyle \frac{1}{{\left({hc}\right)}^{3}}\int \sum _{q}{f}_{0q}{\epsilon }_{0}^{3}d{\epsilon }_{0}{\mu }_{0}^{(0,1,2)}d{{\rm{\Omega }}}_{0} & & = & \displaystyle \frac{4\pi }{{\left({hc}\right)}^{3}}\int \sum _{q}{\psi }_{0}^{(0,1,2)}{\epsilon }_{0}^{3}d{\epsilon }_{0}.\end{array}\end{eqnarray} \tag{ 244 }$$

Transforming ${}^{(\nu )}{{ \mathcal T }}^{\hat{{\rm{a}}}\hat{{\rm{b}}}}$ written in terms of ${E}_{\nu },{F}_{\nu }/c,{P}_{\nu }$ back to the coordinate basis using Equations (208), and adding the stress-energy components of a perfect fluid, given by

$$\begin{eqnarray}&&{}^{(m)}{T}^{\alpha \beta }=(\rho {c}^{2}+{E}_{{\rm{m}}}+{P}_{{\rm{m}}})\displaystyle \frac{{u}^{\alpha }{u}^{\beta }}{{c}^{2}}+{P}_{{\rm{m}}}{g}^{\alpha \beta },\end{eqnarray} \tag{ 245 }$$

with ${u}^{\alpha }$ defined immediately above Equation (209) and ${g}_{\alpha \beta }$ given by Equation (207), the combined matter–neutrino stress energy tensor is given by

$$\begin{eqnarray}{{ \mathcal T }}^{\alpha \beta }=\left(\begin{array}{cccc}\displaystyle \frac{1}{{a}^{2}}\left(\rho {c}^{2}+{E}_{m}+{E}_{\nu }\right) & \displaystyle \frac{1}{{bac}}{F}_{\nu } & 0 & 0\\ \displaystyle \frac{1}{{bac}}{F}_{\nu } & \displaystyle \frac{1}{{b}^{2}}\left({P}_{m}+{P}_{\nu }\right) & 0 & 0\\ 0 & 0 & \displaystyle \frac{1}{{R}^{2}}\left({P}_{m}+\displaystyle \frac{1}{2}\left({E}_{\nu }-{P}_{\nu }\right)\right) & 0\\ 0 & 0 & 0 & \displaystyle \frac{1}{{R}^{2}{\sin }^{2}\theta }\left({P}_{m}+\displaystyle \frac{1}{2}\left({E}_{\nu }-{P}_{\nu }\right)\right)\end{array}\right),\end{eqnarray} \tag{ 246 }$$

where ρ is the proper rest mass density, and E_m and P_m are the matter internal energy density and pressure, respectively.

The Einstein field equations are given by

$$\begin{eqnarray}&&-\displaystyle \frac{8\pi G}{{c}^{4}}\left({T}_{\nu }^{\ \ \mu }\right)={{ \mathcal R }}_{\nu }^{\ \ \mu }-\displaystyle \frac{1}{2}{g}_{\nu }^{\ \ \mu }{{ \mathcal R }}_{\lambda }^{\ \ \lambda },\end{eqnarray} \tag{ 247 }$$

where ${{ \mathcal R }}_{\nu }^{\ \ \mu }$ is the Ricci tensor. After some algebra (e.g., May & White 1966, 1967), we find from the Einstein equations that the gravitational mass is given by

$$\begin{eqnarray}\begin{array}{rclrcl}{M}_{{\rm{g}}} & = & {M}_{{\rm{g}}}(t,R) & & = & {\displaystyle \int }_{0}^{R}\left[\left(\rho +\displaystyle \frac{{E}_{{\rm{m}}}}{{c}^{2}}+\displaystyle \frac{{E}_{\nu }}{{c}^{2}}\right)+\displaystyle \frac{1}{{c}^{4}}\displaystyle \frac{{{uF}}_{\nu }}{{\rm{\Gamma }}}\right]4\pi {R}^{2}{dR},\end{array}\end{eqnarray} \tag{ 248 }$$

and the metric parameter Γ is given by

$$\begin{eqnarray}&&{\rm{\Gamma }}=\sqrt{1+\displaystyle \frac{{u}^{2}}{{c}^{2}}-\displaystyle \frac{2{M}_{{\rm{g}}}G}{{{Rc}}^{2}}},\end{eqnarray} \tag{ 249 }$$

where u is the velocity, $u={a}^{-1}{({dR}/{dt})}_{m}$. Einstein equations can also be used to derive the radial equation of motion, which is given by

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{1}{a}\displaystyle \frac{{du}}{{dt}} & = & -G\displaystyle \frac{{M}_{{\rm{g}}}}{{R}^{2}}-\displaystyle \frac{{{\rm{\Gamma }}}^{2}}{\rho w}\displaystyle \frac{\partial {P}_{{\rm{m}}}}{\partial R}-\displaystyle \frac{4\pi G}{{c}^{2}}R({P}_{{\rm{m}}}+{P}_{\nu })\,-\displaystyle \frac{{\rm{\Gamma }}}{\rho w}\displaystyle \frac{4\pi c}{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{2}d{\epsilon }_{0}\ \sum _{q}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{q}(x,p\right)}_{S}^{(1)},\end{array}\end{eqnarray} \tag{ 250 }$$

where w, the specific enthalpy, is given by

$$\begin{eqnarray}&&w=1+\displaystyle \frac{{E}_{m}+{P}_{m}}{\rho {c}^{2}},\end{eqnarray} \tag{ 251 }$$

and where the sum over ν is a sum over all neutrino and antineutrino species. Equation (250), with $w={\rm{\Gamma }}=a=1$ (Newtonian approximation), and with the centrifugal term added, is the radial Equation (113) and, in differenced form, Equation (114). Here, we consider the neutrino component of these equations; i.e., the last term of Equation (250), which corresponds to the term ${f}_{\nu r}$ in the radial Equation (113). The neutrino contributions to the θ- and ϕ-components of the velocity and energy hydrodynamics equations are described in the text following Equations (113) and in Equations (115), (122), and (123).

Using Equation (199) and the first of Equation (208), the last term of Equation (250) in the Newtonian approximation can be written as

$$\begin{eqnarray}\begin{array}{rcllcl}{f}_{\nu ,r} & = & -\displaystyle \frac{{\rm{\Gamma }}}{\rho w}\displaystyle \frac{4\pi c}{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{2}d{\epsilon }_{0}\ \sum _{q}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{q}(x,p\right)}_{S}^{(1)} & & \to & -\displaystyle \frac{1}{\rho }\displaystyle \frac{4\pi }{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{3}d{\epsilon }_{0}\ \sum _{q}{\left(\displaystyle \frac{1}{c}\displaystyle \frac{d}{{dt}}{f}_{q}(x,p\right)}_{S}^{(1)}.\end{array}\end{eqnarray} \tag{ 252 }$$

The differenced form of the expression given by the last term in Equation (252) for ${f}_{\nu ,r}$ is given in Section 6.9.

To determine the energy–momentum exchange between the matter and neutrinos, we begin with the hydrodynamics equation

$$\begin{eqnarray}&&{}^{(m)}{T}_{\alpha ;\beta }^{\beta }={G}_{\alpha },\end{eqnarray} \tag{ 253 }$$

where, as before, the semicolon denotes covariant differentiation. The left-hand side is the divergence of the stress-energy tensor of the matter, and ${G}^{\alpha }$ is the four-force density—i.e., the negative of the matter to neutrino energy–momentum transfer rate per unit volume. To determine the latter, we operate on Equation (215), or Equation (219), by the negative of the four-momentum density operator (the integral of the product of the invariant momentum volume element and the neutrino four-momentum), which, in the orthonormal basis, is given by

$$\begin{eqnarray}\begin{array}{ll}-\displaystyle \frac{1}{{\left({hc}\right)}^{3}}\int \sum _{q}\displaystyle \frac{{\epsilon }_{0}^{2}d{\epsilon }_{0}d{{\rm{\Omega }}}_{p}}{{\epsilon }_{0}/c}\displaystyle \frac{{\epsilon }_{0}}{c}\left(1,{\mu }_{0},{\left(1-{\mu }_{0}^{2}\right)}^{1/2}\cos {\phi }_{p},\right. & \left.{\left(1-{\mu }_{0}^{2}\right)}^{1/2}\sin {\phi }_{p}\right),\end{array}\end{eqnarray} \tag{ 254 }$$

and use Equations (244) and (212) to get

$$\begin{eqnarray}&&\begin{array}{l}-\displaystyle \frac{1}{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{2}d{\epsilon }_{0}d{{\rm{\Omega }}}_{p}\left(1,{\mu }_{0},{\left(1-{\mu }_{0}^{2}\right)}^{1/2}\cos {\phi }_{p},\,\right.\left.{\left(1-{\mu }_{0}^{2}\right)}^{1/2}\sin {\phi }_{p}\right){\epsilon }_{0}\left\{\displaystyle \frac{1}{a}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial t}\right)}_{m,{\epsilon }_{0},{\mu }_{0}}\cdots \right\}\\ \quad =\,\left[-\displaystyle \frac{1}{a}{\left(\displaystyle \frac{\partial {E}_{\nu }}{\partial t}\right)}_{m,{E}_{0},{\mu }_{0}}-\,\cdots ,\,-\displaystyle \frac{1}{{ca}}{\left(\displaystyle \frac{\partial {F}_{\nu }}{\partial t}\right)}_{m,{E}_{0},{\mu }_{0}}\right.\left.-\,\cdots ,\,0-\cdots ,\,0-\cdots \Space{0ex}{3.8ex}{0ex}\right]\\ \quad =\,-\displaystyle \frac{4\pi c}{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{2}d{\epsilon }_{0}\ \left(\sum _{q}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0q}\right)}_{S}^{(0)},\,\sum _{q}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0q}\right)}_{S}^{(1)},0,0\right)\\ \quad =\,\left({G}^{\hat{1}},{G}^{\hat{2}},{G}^{\hat{3}},{G}^{\hat{4}}\right).\end{array}\end{eqnarray} \tag{ 255 }$$

Transforming Equation (255) to the coordinate basis, we have for ${G}^{\alpha }$,

$$\begin{eqnarray}\begin{array}{rclrcl}{G}^{\alpha } & = & -\displaystyle \frac{4\pi c}{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{2}d{\epsilon }_{0}\ \left(\displaystyle \frac{1}{a}\sum _{q}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0q}\right)}_{S}^{(0)},\right. & & & \left.\displaystyle \frac{1}{b}\sum _{q}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0q}\right)}_{S}^{(1)},0,0\right).\end{array}\end{eqnarray} \tag{ 256 }$$

The energy equation is obtained by projecting Equation (253) along the fluid four-velocity:

$$\begin{eqnarray}&&{u}^{\alpha (m)}{T}_{\alpha ;\beta }^{\beta }={u}^{\alpha }{G}_{\alpha }=-\displaystyle \frac{4\pi {c}^{2}}{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{2}d{\epsilon }_{0}\ \sum _{q}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0q}\right)}_{S}^{(0)},\end{eqnarray} \tag{ 257 }$$

where we have used Equation (256) for ${G}^{\alpha }$. Using Equation (245) for ${}^{(m)}{T}^{\alpha \beta }$ on the left-hand side of Equation (253), invoking conservation of rest mass $[{u}_{;\beta }^{\beta }=-{u}^{\beta }{\rho }_{,\beta }/\rho$], and denoting by ${e}_{{\rm{m}}}$ the internal energy per unit rest mass (${e}_{{\rm{m}}}={E}_{{\rm{m}}}/\rho$), ${u}^{\alpha (m)}{T}_{\alpha ;\beta }^{\beta }$ is given by

$$\begin{eqnarray}&&{u}^{\alpha (m)}{T}_{\alpha ;\beta }^{\beta }=\rho \left[\displaystyle \frac{1}{a}\displaystyle \frac{\partial {e}_{m}}{\partial t}+{P}_{m}\displaystyle \frac{1}{a}\displaystyle \frac{\partial (1/\rho )}{\partial t}\right],\end{eqnarray} \tag{ 258 }$$

and Equation (257) becomes

$$\begin{eqnarray}\begin{array}{rcll}\displaystyle \frac{1}{a}{\left(\displaystyle \frac{\partial {e}_{{\rm{m}}}}{\partial t}\right)}_{m} & = & -{p}_{{\rm{m}}}{\left(\displaystyle \frac{1}{a}\displaystyle \frac{\partial \left(1/\rho \right)}{\partial t}\right)}_{m}-\displaystyle \frac{4\pi {c}^{2}}{{\left({hc}\right)}^{3}}\displaystyle \frac{1}{\rho } & \displaystyle \int {\epsilon }_{0}^{2}d{\epsilon }_{0}\ \sum _{q}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0q}\right)}_{S}^{(0)},\end{array}\end{eqnarray} \tag{ 259 }$$

which states that changes in the matter internal energy arise from work by local compression or expansion and from energy exchange with neutrinos.

The other nontrivial equation in (253) is the radial equation

$$\begin{eqnarray}&&{}^{(m)}{T}_{\ \ \ ;\beta }^{1\beta }={G}^{1}.\end{eqnarray} \tag{ 260 }$$

With ${}^{(m)}{T}_{\ \ \ ;\beta }^{1\beta }$ given by

$$\begin{eqnarray}\begin{array}{rclrcl}{}^{(m)}{T}_{\ \ \ ;\beta }^{1\beta } & = & \displaystyle \frac{1}{{b}^{2}}\left[\displaystyle \frac{\partial \mathrm{ln}a}{\partial m}\left(\rho {c}^{2}+{E}_{m}+{P}_{m}\right)+\displaystyle \frac{\partial {P}_{m}}{\partial m}\right] & & = & \displaystyle \frac{{\rm{\Gamma }}}{b}\left[\displaystyle \frac{\partial \mathrm{ln}a}{\partial R}\left(\rho {c}^{2}+{E}_{m}+{P}_{m}\right)+\displaystyle \frac{\partial {P}_{m}}{\partial R}\right],\end{array}\end{eqnarray} \tag{ 261 }$$

Equation (260) gives

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{ln}a}{{dR}}=-\left(\displaystyle \frac{{{dP}}_{m}}{{dR}}+\displaystyle \frac{4\pi c}{{\rm{\Gamma }}{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{2}d{\epsilon }_{0}\ \sum _{q}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0q}\right)}_{S}^{(1)}\right)\displaystyle \frac{1}{\rho {{wc}}^{2}},\end{eqnarray} \tag{ 262 }$$

where w is given by Equation (251).

Equation (262) relates the spatial gradient of the matter pressure, ${P}_{{\rm{m}}}$, and the rate of neutrino–matter momentum exchange to the spatial gradient of the lapse function. Upon integrating a from the surface using Equation (262), we start with the surface (subscript s) boundary condition, which we take to be the Schwarzschild solution on the exterior, and neglect the small deviations from this induced by the radiation; namely,

$$\begin{eqnarray}&&{a}_{s}=\left(1-\displaystyle \frac{2{{GM}}_{s}}{{c}^{2}{R}_{s}}\right)\displaystyle \frac{1}{{{\rm{\Gamma }}}_{s}},\end{eqnarray} \tag{ 263 }$$

so that the coordinate time is that of a distant clock. In practice, our models are extended enough that a_s is extremely close to unity. The differencing of the lapse is given below by Equations (294) and (295).

The local lepton number density of the matter, ${}^{(m)}{n}_{{\ell }}$, is given from charge conservation by

$$\begin{eqnarray}&&{}^{(m)}{n}_{{\ell }}={n}_{{\rm{p}}}={n}_{{{\rm{e}}}^{-}}-{n}_{{{\rm{e}}}^{+}}=\displaystyle \frac{\rho }{{m}_{{\rm{B}}}}{Y}_{{\rm{e}}},\end{eqnarray} \tag{ 264 }$$

where ${n}_{{\rm{p}}}$, ${n}_{{\rm{B}}}$, ${n}_{{{\rm{e}}}^{-}}$, and ${n}_{{{\rm{e}}}^{+}}$ are the proper proton, baryon, electron, and positron number densities, respectively, ${Y}_{{\rm{e}}}$ is the proton fraction, and ${m}_{{\rm{B}}}$ is the mean baryon mass. Referring to Equation (214), the local lepton number density in neutrinos, ${}^{(\nu )}{n}_{{\ell }}$, is given by

$$\begin{eqnarray}&&{}^{(\nu )}{n}_{{\ell }}=\displaystyle \frac{1}{{\left({hc}\right)}^{3}}\int \left({f}_{0{\nu }_{e}}-{f}_{0{\bar{\nu }}_{e}}\right){\epsilon }_{0}^{2}d{\epsilon }_{0}d{{\rm{\Omega }}}_{p}.\end{eqnarray} \tag{ 265 }$$

Transport produces global changes in ${}^{(\nu )}{n}_{{\ell }}$, which, by itself, does not change ${}^{(m)}{n}_{{\ell }}$ (or ${Y}_{{\rm{e}}}$). Ignoring the effects of transport, the total lepton number, ${}^{\mathrm{total}}{n}_{{\ell }}{=}^{(m)}{n}_{{\ell }}{+}^{(\nu )}{n}_{{\ell }}$, is a locally conserved quantity, and this allows us to relate the change in ${Y}_{{\rm{e}}}$ to the change in ${}^{(\nu )}{n}_{{\ell }}$. In particular,

$$\begin{eqnarray}&&0={\left({}^{\mathrm{total}}{n}_{{\ell }}{u}^{\alpha }\right)}_{;\alpha }={\left(\displaystyle \frac{\rho }{{m}_{{\rm{B}}}}{Y}_{{\rm{e}}}{u}^{\alpha }\right)}_{;\alpha }+{\left({}^{(\nu )}{n}_{{\ell }}{u}^{\alpha }\right)}_{;\alpha }.\end{eqnarray} \tag{ 266 }$$

Expanding the covariant derivative of ${Y}_{{\rm{e}}}$ with the use of Equation (210), which expresses rest mass conservation, Equation (266) can be written

$$\begin{eqnarray}&&{\left(\displaystyle \frac{\rho }{{m}_{{\rm{B}}}}{Y}_{{\rm{e}}}{u}^{\alpha }\right)}_{;\alpha }=\displaystyle \frac{\rho }{{m}_{{\rm{B}}}}\displaystyle \frac{1}{a}{\left(\displaystyle \frac{\partial {Y}_{{\rm{e}}}}{\partial t}\right)}_{m}=-{\left({}^{(\nu )}{n}_{{\ell }}{u}^{\alpha }\right)}_{;\alpha }.\end{eqnarray} \tag{ 267 }$$

Now expanding the covariant derivative of ${}^{(\nu )}{n}_{{\ell }}$, Equation (267) becomes

$$\begin{eqnarray}&&\displaystyle \frac{\rho }{{m}_{{\rm{B}}}}\displaystyle \frac{1}{a}{\left(\displaystyle \frac{\partial {Y}_{{\rm{e}}}}{\partial t}\right)}_{m}=-\displaystyle \frac{\rho }{a}{\left(\displaystyle \frac{\partial \left({}^{(\nu )}{n}_{{\ell }}/\rho \right)}{\partial t}\right)}_{m}.\end{eqnarray} \tag{ 268 }$$

With the use of Equations (215) (without the transport terms), (199), (211), and (214), the term on the right-hand side of Equation (268) becomes

$$\begin{eqnarray}&&\begin{array}{l}-\displaystyle \frac{\rho }{a}{\left(\displaystyle \frac{\partial \left({}^{(\nu )}{n}_{{\ell }}/\rho \right)}{\partial t}\right)}_{m}=\,-\displaystyle \frac{1}{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{2}d{\epsilon }_{0}d{{\rm{\Omega }}}_{p}\left(\displaystyle \frac{1}{a}{\left(\displaystyle \frac{\partial {f}_{0{\nu }_{e}}}{\partial t}\right)}_{m}-\displaystyle \frac{1}{a}{\left(\displaystyle \frac{\partial {f}_{0{\bar{\nu }}_{e}}}{\partial t}\right)}_{m}\right)\\ \quad =\,-\displaystyle \frac{1}{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{2}d{\epsilon }_{0}d{{\rm{\Omega }}}_{p}\displaystyle \frac{{c}^{2}}{{\epsilon }_{0}}\left[{\left(\displaystyle \frac{{{df}}_{0{\nu }_{e}}}{d{\ell }}\right)}_{S}-{\left(\displaystyle \frac{{{df}}_{0{\bar{\nu }}_{e}}}{d{\ell }}\right)}_{S}\right]\\ \quad =\,\displaystyle \frac{c}{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{2}d{\epsilon }_{0}\left[\left({j}_{{\nu }_{e}}({\epsilon }_{0})\left(1-{\psi }_{0\,{\nu }_{e}}^{(0)}({\epsilon }_{0}\right)-\displaystyle \frac{{\psi }_{0\,{\nu }_{e}}^{(0)}({\epsilon }_{0})}{{\lambda }_{{\nu }_{{\rm{e}}}}^{a}({\epsilon }_{0})}\right)\right.\\ \qquad \left.-\,\left({j}_{{\bar{\nu }}_{{\rm{e}}}}({\epsilon }_{0})\left(1-{\psi }_{0\,{\bar{\nu }}_{e}}^{(0)}({\epsilon }_{0}\right)-\displaystyle \frac{{\psi }_{0\,{\bar{\nu }}_{e}}^{(0)}({\epsilon }_{0})}{{\lambda }_{{\bar{\nu }}_{e}}^{a}({\epsilon }_{0})}\right)\right],\end{array}\end{eqnarray} \tag{ 269 }$$

where the last expression in Equation (269) arises from the fact that the only terms that do not vanish upon integrating over solid angle and neutrino number are emission and absorption, and the latter are isotropic. Here, ${j}_{{\nu }_{{\rm{e}}}}({\epsilon }_{0})$ and $1/{\lambda }_{{\nu }_{e}}^{a}({\epsilon }_{0})$ are the electron-neutrino emission and absorption per state per unit length, and ${j}_{{\bar{\nu }}_{e}}({\epsilon }_{0})$ and $1/{\lambda }_{{\bar{\nu }}_{e}}^{a}({\epsilon }_{0})$ are the electron–antineutrino counterparts. Equations (266)–(269) give the following equation for the evolution of the electron fraction due to sources:

$$\begin{eqnarray}\begin{array}{ll}{\left(\displaystyle \frac{\partial {Y}_{{\rm{e}}}}{\partial t}\right)}_{m}=-\displaystyle \frac{4\pi {ac}}{{\left({hc}\right)}^{3}}\,\displaystyle \frac{{m}_{{\rm{B}}}}{\rho }\int {\epsilon }_{0}^{2}d{\epsilon }_{0}\left[\Space{0ex}{3.7ex}{0ex}\left(\Space{0ex}{3.7ex}{0ex}{j}_{{\nu }_{{\rm{e}}}}({\epsilon }_{0})\left(1-{\psi }_{0\,{\nu }_{e}}^{(0)}({\epsilon }_{0},)\right)\right.\right. & \left.\left.-\displaystyle \frac{{\psi }_{0\,{\nu }_{e}}^{(0)}({\epsilon }_{0})}{{\lambda }_{{\nu }_{e}}({\epsilon }_{0})}\right)-\left({j}_{{\bar{\nu }}_{e}}({\epsilon }_{0})\left(1-{\psi }_{0\,{\bar{\nu }}_{e}}^{(0)}({\epsilon }_{0}\right)-\displaystyle \frac{{\psi }_{0\,{\bar{\nu }}_{e}}^{(0)}({\epsilon }_{0})}{{\lambda }_{{\bar{\nu }}_{{\rm{e}}}}({\epsilon }_{0})}\right)\right].\end{array}\end{eqnarray} \tag{ 270 }$$

In the Lagrangian transport step, the pair of transport Equations (240) and (229) (or, rather, Equation (272) in place of Equation (240), where Equation (272) is Equation (240) written in conservative form) are solved along with the neutrino-specific term of the energy Equation (259), namely,

$$\begin{eqnarray}&&\displaystyle \frac{1}{a}{\left(\displaystyle \frac{\partial {e}_{{\rm{m}}}}{\partial t}\right)}_{m}=-\displaystyle \frac{4\pi {c}^{2}}{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{2}d{\epsilon }_{0}\ \sum _{q}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0q}\right)}_{S}^{(0)},\end{eqnarray} \tag{ 271 }$$

and Equation (270) for the change in the matter proton number. The pair of Equations (240) and (229) for each neutrino species and Equation (270) are solved together fully implicitly, while Equation (271) is solved immediately afterwards, as described in more detail below. It was found that executing Equation (271) subsequent to the others leads to a substantially better conditioned set of equations and no loss of stability and that the internal energy update could then be easily omitted for those radial zones for which the internal energy update is derived from a total energy update. The role of neutrinos in the latter case will be described later. In either case, the updated temperature is obtained from the updated internal energy and electron fraction by numerically inverting the equation of state.

Equation (240), written in conservative form, is given by

$$\begin{eqnarray}\begin{array}{ll}\displaystyle \frac{1}{{ac}}{\left(\displaystyle \frac{\partial {\psi }_{0\,q}^{(0)}}{\partial t}\right)}_{m,a\epsilon }^{}+{\rm{\Gamma }}\displaystyle \frac{{a}^{2}}{{R}^{2}}{\left(\displaystyle \frac{\partial }{\partial R}\left(\displaystyle \frac{{R}^{2}}{{a}^{2}}{\psi }_{0q}^{(1)}\right)\right)}_{t,e} & \,=\,\displaystyle \frac{c}{{\epsilon }_{0}}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0q}(x,p\right)}_{S}^{(0)},\end{array}\end{eqnarray} \tag{ 272 }$$

where we have omitted the superscript T. Consider a radial mesh where, as before, the zone edges are labeled by $i+{}^{1}{/}_{2}$ and the zone centers by i. Consider also an energy mesh labeled similarly by k. Let the subscript q denote, as before, the neutrino species, and let the superscript n denote the nth time slice. Then, for each neutrino species, Equation (272) is differenced conservatively as follows:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{{a}_{i}c}\displaystyle \frac{{\psi }_{0\,i,k,q}^{(0)\,n+1}-{\psi }_{0\,i,k,q}^{(0)\,n}}{{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}}\\ \qquad +\displaystyle \frac{{a}_{i}^{2}}{{\mathrm{Vol}}_{i}}\left[\displaystyle \frac{1}{{a}_{i+\tfrac{1}{2}}^{2}}{\mathrm{Area}}_{i+\displaystyle \frac{1}{2}}{\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}\right.\left.-\,\displaystyle \frac{1}{{a}_{i-\tfrac{1}{2}}^{2}}{\mathrm{Area}}_{i-\displaystyle \frac{1}{2}}{\psi }_{0\,i-\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}\right] & =\,\displaystyle \frac{c}{{\epsilon }_{0}}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0}(x,p\right)}_{S\ i,k,q}^{(0)\,n+1},\end{array}\end{eqnarray} \tag{ 273 }$$

with ${\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}$ given by

$$\begin{eqnarray}&&{\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}=-\displaystyle \frac{{\lambda }_{i+\tfrac{1}{2},k,q}^{(t)\,n+1}}{3}{{ \mathcal F }}_{i+\tfrac{1}{2},k,q}^{n+1}{{\rm{\Gamma }}}_{i+\tfrac{1}{2}}^{n+1}\displaystyle \frac{{\psi }_{0\,i+1,k,q}^{(0)\,n+1}-{\psi }_{0\,i,k,q}^{(0)\,n+1}}{{R}_{i+1}-{R}_{i}},\end{eqnarray} \tag{ 274 }$$

(or by its modification given by Equation (300) below), where ${\mathrm{Area}}_{i+\tfrac{1}{2}}=4\pi {R}_{i+\tfrac{1}{2}}^{2}$ is the proper area of the outer boundary of zone i, ${R}_{i}=({R}_{i-\tfrac{1}{2}}+{R}_{i+\tfrac{1}{2}})/2$, ${\mathrm{Vol}}_{i}=4\pi \left({R}_{i+\tfrac{1}{2}}^{3}-{R}_{i-\tfrac{1}{2}}^{3}\right)/3{\rm{\Gamma }}$ is the proper volume enclosed between zones edges $i-{}^{1}{/}_{2}$ and $i+{}^{1}{/}_{2}$, ${\rm{\Delta }}{t}^{n+\tfrac{1}{2}}={t}^{n+1}-{t}^{n}$, and ${\lambda }_{i+\tfrac{1}{2},k,m}^{(t)\,n+1}=\sqrt{{\lambda }_{i,k,m}^{(t)\,n+1}{\lambda }_{i+1,k,m}^{(t)\,n+1}}$. To demonstrate that Equation (273) is conservative in neutrino number, first multiply it by $c\,{\mathrm{Vol}}_{i}\,{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}/{a}_{i}^{2}$ to rewrite it as

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{{a}_{i}^{3}}\left({\psi }_{0\,i,k,q}^{(0)\,n+1}-{\psi }_{0\,i,k,q}^{(0)\,n}\right){\mathrm{Vol}}_{i}=\,-\displaystyle \frac{1}{{a}_{i+\tfrac{1}{2}}^{3}}{\mathrm{Area}}_{i+\displaystyle \frac{1}{2}}{\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}{a}_{i+\tfrac{1}{2}}{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}c\,+\,\displaystyle \frac{1}{{a}_{i-\tfrac{1}{2}}^{3}}{\mathrm{Area}}_{i-\displaystyle \frac{1}{2}}{\psi }_{0\,i-\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}{a}_{i-\tfrac{1}{2}}{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}c\\ \qquad +\,{a}_{i}{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}\displaystyle \frac{c}{{a}_{i}^{3}}{\mathrm{Vol}}_{i}\times \displaystyle \frac{c}{{\epsilon }_{0}}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0}(x,p\right)}_{S\ i,k}^{(0)n+1}.\end{array}\end{eqnarray} \tag{ 275 }$$

According to the discussion following Equation (219) and our choice of the lapse function limit, $a(R,t)\to 1$ as $R\to \infty$, a neutrino propagating along a geodesic passing through ${R}_{i-\tfrac{1}{2}}$, R_i, and ${R}_{i+\tfrac{1}{2}}$ in a static spacetime will have locally measured energies at these points related by

$$\begin{eqnarray}&&{\epsilon }_{0i-\displaystyle \frac{1}{2}}{a}_{i-\tfrac{1}{2}}={\epsilon }_{0i}{a}_{i}={\epsilon }_{0i+\displaystyle \frac{1}{2}}{a}_{i+\tfrac{1}{2}}={\epsilon }_{0\infty }.\end{eqnarray} \tag{ 276 }$$

Thus, multiplying Equation (275) by ${\left({hc}\right)}^{-3}{\left({\epsilon }_{0k\infty }\right)}^{2}{\rm{\Delta }}{\epsilon }_{0k\infty }$ and using Equation (276), we get

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\left({\epsilon }_{0i,k}\right)}^{2}{\rm{\Delta }}{\epsilon }_{0i,k}}{{\left({hc}\right)}^{3}}\left({\psi }_{0\,i,k,q}^{(0)\,n+1}-{\psi }_{0\,i,k,q}^{(0)\,n}\right){\mathrm{Vol}}_{i}\,=\,-{\mathrm{Area}}_{i+\displaystyle \frac{1}{2}}\displaystyle \frac{c\,{\left({\epsilon }_{0i+\tfrac{1}{2},k}\right)}^{2}{\rm{\Delta }}{\epsilon }_{0i+\tfrac{1}{2},k}}{{\left({hc}\right)}^{3}}{\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}{a}_{i+\tfrac{1}{2}}{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}\\ \qquad +\,{\mathrm{Area}}_{i-\displaystyle \frac{1}{2}}\displaystyle \frac{c\,{\left({\epsilon }_{0i-\tfrac{1}{2},k}\right)}^{2}{\rm{\Delta }}{\epsilon }_{0i-\tfrac{1}{2},k}}{{\left({hc}\right)}^{3}}{\psi }_{0\,i-\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}{a}_{i-\tfrac{1}{2}}{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}\,+\,{a}_{i}{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}\displaystyle \frac{{\left({\epsilon }_{0i,k}\right)}^{2}{\rm{\Delta }}{\epsilon }_{0i,k}}{{\left({hc}\right)}^{3}}{\mathrm{Vol}}_{i}\times {\left(\displaystyle \frac{d}{{dt}}{f}_{0}\right)}_{S\ i,k}^{(0)n+1}.\end{array}\end{eqnarray} \tag{ 277 }$$

The left-hand side of Equation (277) is the change in the number of neutrinos in ${\mathrm{Vol}}_{i}$ between energies ${\epsilon }_{0i,k}$ and ${\epsilon }_{0i,k}+{\rm{\Delta }}{\epsilon }_{0i,k}$ during the proper time interval ${a}_{i}{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}$. The first term on the right-hand side subtracts the number of neutrinos that leave through the outer surface of ${\mathrm{Vol}}_{i}$ between energies ${\epsilon }_{0i+\tfrac{1}{2},k}$ and ${\epsilon }_{0i+\tfrac{1}{2},k}+{\rm{\Delta }}{\epsilon }_{0i+\tfrac{1}{2},k}$ in proper time ${a}_{i+\tfrac{1}{2}}{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}$. The second term on the right-hand side adds the number of neutrinos that enter through the inner surface of ${\mathrm{Vol}}_{i}$ between energies ${\epsilon }_{0i-\tfrac{1}{2},k}$ and ${\epsilon }_{0i-\tfrac{1}{2},k}+{\rm{\Delta }}{\epsilon }_{0i-\tfrac{1}{2},k}$ in proper time ${a}_{i-\tfrac{1}{2}}{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}$. The last term is the net number of neutrinos emitted, absorbed, or scattered in/out of ${\mathrm{Vol}}_{i}$ between energies ${\epsilon }_{0i,k}$ and ${\epsilon }_{0i,k}+{\rm{\Delta }}{\epsilon }_{0i,k}$ in proper time ${a}_{i}{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}$.

Equation (270) is differenced straightforwardly as

$$\begin{eqnarray}\displaystyle \begin{array}{rcl}\displaystyle \frac{1}{{a}_{i}c}\displaystyle \frac{{Y}_{{\rm{e}},i}^{n+1}-{Y}_{{\rm{e}},i}^{n}}{{\rm{\Delta }}t} & = & -\displaystyle \frac{4\pi }{{\left({hc}\right)}^{3}}\,\displaystyle \frac{{m}_{{\rm{B}}}}{{\rho }_{i}}\sum _{k=1}^{{N}_{k}}{\epsilon }_{0\,i,k}^{2}{\rm{\Delta }}{\epsilon }_{0i,k}\\ & & \times \left[\left({j}_{{\nu }_{e}\,i,k}^{n+1}\left(1-{\psi }_{0\,i,k,{\nu }_{e}}^{(0)\,n+1}\right)-\displaystyle \frac{{\psi }_{0\,i,k,{\nu }_{e}}^{(0)\,n+1}}{{\lambda }_{{\nu }_{e}\,i,k}^{n+1}}\right)\right. & & & \left.-\left({j}_{{\bar{\nu }}_{e}\,i,k}^{n+1}\left(1-{\bar{\psi }}_{0\,i,k,{\nu }_{e}}^{(0)\,n+1}\right)-\displaystyle \frac{{\bar{\psi }}_{0\,i,k,{\nu }_{e}}^{(0)\,n+1}}{{\lambda }_{{\bar{\nu }}_{e}\,i,k}^{n+1}}\right)\right].\end{array}\end{eqnarray} \tag{ 278 }$$

During the Lagrangian transport step, the set of Equations (273) and (274), modified in the presence of shocks as given by Equations (300), (306), and (278) are solved implicitly for ${Y}_{{\rm{e}},i}^{n+1}$ and ${\psi }_{0\,i,k,q}^{(0)\,n+1}$ for each neutrino species q.

The temperature change accompanying the Lagrangian transport step is computed following the solutions of Equations (273), (274), and (278), from the change in the internal energy as given by the following differenced version of Equation (271):

$$\begin{eqnarray}\displaystyle \begin{array}{rclrcl}\displaystyle \frac{1}{{a}_{i}c}\displaystyle \frac{{e}_{\mathrm{int},i}^{n+1}-{e}_{\mathrm{int},i}^{n}}{{\rm{\Delta }}t} & = & -\displaystyle \frac{4\pi }{{\left({hc}\right)}^{3}}\,\displaystyle \frac{{m}_{{\rm{B}}}}{{\rho }_{i}}\sum _{q=1}^{{N}_{\nu }}{{ \mathcal N }}_{\mathrm{stwt},q} & & & \sum _{k=1}^{{N}_{k}}{\epsilon }_{0\,i,k}^{3}{\rm{\Delta }}{\epsilon }_{0i,k}\displaystyle \frac{c}{{\epsilon }_{0}}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0}(x,p\right)}_{S\,i,k,q}^{(0)\,n+1},\end{array}\end{eqnarray} \tag{ 279 }$$

and the change in ${Y}_{{\rm{e}}}$ given by Equation (278), where ${N}_{\nu }$ is the number of neutrino species (typically four), and ${{ \mathcal N }}_{\mathrm{stwt},q}$ is the statistical weight of each neutrino species, q, (typically 1 for ${\nu }_{e}$ and ${\bar{\nu }}_{e}$, and 2 for ${\nu }_{\mu \tau }$ and ${\bar{\nu }}_{\mu \tau }$). Specifically,

$$\begin{eqnarray}&&{T}_{i}^{n+1}={T}_{i}^{n}+\displaystyle \frac{{e}_{\mathrm{int},i}^{n+1}-{e}_{\mathrm{int},i}^{n}+\left({Y}_{{\rm{e}},i}^{n+1}-{Y}_{{\rm{e}},i}^{n}\right){\left(\tfrac{{{de}}_{\mathrm{int}}}{{{dY}}_{{\rm{e}}}}\right)}_{\rho ,{Y}_{{\rm{e}}}\ i}^{n+\tfrac{1}{2}}}{{\left(\tfrac{{{de}}_{\mathrm{int}}}{{dT}}\right)}_{\rho ,{Y}_{{\rm{e}}}\ i}^{n+\tfrac{1}{2}}}.\end{eqnarray} \tag{ 280 }$$

It was found that the temperature change during a time step always had very little effect on the terms in the transport equation, and therefore on the stability of the difference scheme. On the other hand, including the temperature change as part of the implicit solution of Equations (273), (274), and (278) caused the system of equations to be rather ill-conditioned. It was therefore deemed numerically expedient to solve for the temperature change after, rather than simultaneously with, the solution of the transport equations. The solution of Equations (273), (274), and (278) followed by Equation (279) and (280) completes the Lagrangian transport step.

The numerical solution of the set of transport equations from time step n to time step $n+1$ proceeds by an outer iteration for the corrections $\delta {\psi }_{0\,i,k,q}^{(0)\,n,i+1}$ and $\delta {Y}_{{\rm{e}},i}^{n,i+1}$ to the quantities ${\psi }_{0\,i,k,q}^{(0)\,n,i}$ and $\delta {Y}_{{\rm{e}},i}^{n,i};$ i.e.,

$$\begin{eqnarray}&&{\psi }_{0\,i,k,q}^{(0)\,n,i+1}={\psi }_{0\,i,k,q}^{(0)\,n,i}+\delta {\psi }_{0\,i,k,q}^{(0)\,n,i+1},\qquad {Y}_{{\rm{e}},i}^{n,i+1}={Y}_{{\rm{e}},i}^{n,i}+\delta {Y}_{{\rm{e}},i}^{n,i+1}\end{eqnarray} \tag{ 281 }$$

where the superscript "i" denotes the quantities after the ith iteration. When the corrections have become sufficiently small, the quantities ${\psi }_{0\,i,k,q}^{(0)\,n,i+1}$ and ${Y}_{{\rm{e}},i}^{n,i+1}$ are regarded as having converged to ${\psi }_{0\,i,k,q}^{(0)\,n+1}$ and ${Y}_{{\rm{e}},i}^{n+1}$, respectively. The criterion for convergence is that

$$\begin{eqnarray}\begin{array}{ll}\delta {\psi }_{0\,i,k,q}^{(0)\,n,i+1}\lt \displaystyle \frac{{\mathrm{tol}}_{\psi }}{\max ({\psi }_{0\,i,k,q}^{(0)\,n,i+1},{\mathrm{tolmin}}_{\psi })}, & \,\delta {Y}_{{\rm{e}},i}^{n,i+1}\lt \displaystyle \frac{{\mathrm{tol}}_{{Y}_{{\rm{e}}}}}{\max ({Y}_{{\rm{e}},i}^{n,i+1},{\mathrm{tolmin}}_{{Y}_{{\rm{e}}}})}\end{array}\end{eqnarray} \tag{ 282 }$$

for all ${\psi }_{0\,i,k,q}^{(0)\,n,i+1}$ and ${Y}_{{\rm{e}},i}^{n,i+1}$, where typically ${\mathrm{tol}}_{\psi }={10}^{-6}$, ${\mathrm{tol}}_{{Y}_{{\rm{e}}}}={10}^{-6}$, ${\mathrm{tolmin}}_{\psi }=1$, and ${\mathrm{tolmin}}_{{Y}_{{\rm{e}}}}=0.1$.

The transport equations, Equations (273) and (274), when linearized comprise a set ${N}_{\nu }\times {N}_{k}$ coupled linear equations of the form:

$$\begin{eqnarray}&&\displaystyle \begin{array}{l}\mathrm{LHS}{0}_{i,{n}_{q}+k}^{i}+\mathrm{LHS}{0}_{{\rm{p}}0{\rm{p}}\,i,{n}_{q}+k}^{i}\delta {\psi }_{0\,i,k,q}^{(0)\,n,i+1}+\,\mathrm{LHS}{0}_{{\rm{p}}0{\rm{m}}\,i,{n}_{q}+k}^{i}\delta {\psi }_{0\,i-1,k,q}^{(0)\,n,i+1}+\,\mathrm{LHS}{0}_{{\rm{p}}0\,i,{n}_{q}+k}^{i}\delta {\psi }_{0\,i+1,k,q}^{(0)\,n,i+1}\\ \quad =\,\mathrm{RHS}{0}_{i,{n}_{q}+k}^{i}+\mathrm{RHS}{0}_{{\rm{y}}\,i,{n}_{q}+k}^{i}\delta {Y}_{{\rm{e}},i}^{n,\ i+1}+\,\sum _{q^{\prime} =1}^{{N}_{\nu }}\sum _{k^{\prime} =1}^{{N}_{k}}\mathrm{RHS}{0}_{{\rm{p}}0\,i,{n}_{q}+k,{n}_{{q}^{{\prime} }}^{{\prime} }+k^{\prime} }^{i}\delta {\psi }_{0\,i,k^{\prime} ,q^{\prime} }^{(0)\,n,i+1},\end{array}\end{eqnarray} \tag{ 283 }$$

where the coefficients "LHS0" and "RHS0" are independent of the corrections, and where ${n}_{q}=({N}_{\nu }-1)\times q$, and the notation "LHS0" and "RHS0" denote the left-hand side (transport) and right-hand side (sources) of the transport equations, respectively. Equation (278) provides an additional equation, which is coupled to the preceding set of equations, and when linearized is of the form

$$\begin{eqnarray}&&\displaystyle \begin{array}{l}\mathrm{LHS}{0}_{i}^{\mathrm{Ye}\,i}+\mathrm{LHS}{0}_{{\rm{y}}\,i}^{\mathrm{Ye}\,i}\delta {Y}_{{\rm{e}},i}^{n,i+1}=\,\mathrm{RHS}{0}_{i}^{\mathrm{Ye}\,i}+\mathrm{RHS}{0}_{{\rm{y}}\,i}^{\mathrm{Ye}\,i}\delta {Y}_{{\rm{e}},i}^{n,\ i+1}\\ \qquad +\,\sum _{k=1}^{{N}_{{\rm{e}}}}\mathrm{RHS}{0}_{{\rm{p}}0\,i,k,q=1}^{\mathrm{Ye}i}\delta {\psi }_{0\,i,k,q=1}^{(0)\,n,i+1}+\,\sum _{k=1}^{{N}_{{\rm{e}}}}\mathrm{RHS}{0}_{{\rm{p}}0\,i,k,q=2}^{\mathrm{Ye}i}\delta {\psi }_{0\,i,k,q=2}^{(0)\,n,i+1},\end{array}\end{eqnarray} \tag{ 284 }$$

where q = 1 and q = 2 refer to ${\nu }_{e}$ and ${\bar{\nu }}_{e}$, respectively. Equations (283) and (284) are combined to form a matrix equation for the corrections, of the form

$$\begin{eqnarray}\displaystyle \begin{array}{ll}\sum _{\alpha ^{\prime} =1}^{{N}_{\nu }\times {N}_{k}+1}{A}_{i,\alpha ,\alpha ^{\prime} }\delta {u}_{i,\alpha ^{\prime} }+{{Bm}}_{i,\alpha }\delta {u}_{i-1,\alpha }+{{Bp}}_{i,\alpha }\delta {u}_{i+1,\alpha }+{C}_{i,\alpha }, & i=1,\cdots ,I,\ \alpha =1,\ldots ,{N}_{\nu }\times {N}_{k}+1,\end{array}\end{eqnarray} \tag{ 285 }$$

where the corrections $\delta {\psi }_{0\,i,k,q}^{(0)\,n,i+1}$ and $\delta {Y}_{{\rm{e}},i}^{n,\ i+1}$ are incorporated in the array $\delta {u}_{i,\alpha },\alpha =1,\cdots ,\,{N}_{\nu }\times {N}_{k}+1$.

The boundary condition at the center is simply that the neutrino flux is zero; i.e., ${\psi }_{0\,i=1,k,q}^{(1)\,}=0$. Constant luminosity is assumed at the surface, which requires that ${\psi }_{0\,I+1,k,q}^{(1)\,}{R}_{I+1}^{2}={\psi }_{0\,I,k,q}^{(1)\,}{R}_{I}^{2}$. Following the discussion immediately above Equation (233), a geometric relation between ${\psi }_{0\,I,k,q}^{(1)\,}$ and ${\psi }_{0\,I,k,q}^{(0)\,}$ of the form ${\psi }_{0\,I,k,q}^{(1)\,}=(1/2)(1+{\mu }_{0I,k,q}){\psi }_{0\,I,k,q}^{(0)\,}$ is assumed. Then, without velocity or GR corrections to ${\mu }_{0I,k,q}$, the surface boundary condition for ${\psi }_{0\,I,k,q}^{(0)\,}$ becomes

$$\begin{eqnarray}&&{\psi }_{0\,I+1,k,q}^{(0)\,}={\psi }_{0\,I,k,q}^{(0)\,}\displaystyle \frac{{R}_{I}^{2}-\tfrac{1}{4}{R}_{\nu k,q}}{{R}_{I+1}^{2}-\tfrac{1}{4}{R}_{\nu k,q}},\end{eqnarray} \tag{ 286 }$$

where ${R}_{\nu k,q}$ is the neutrinosphere radius for neutrinos of energy k and type q.

Once ${\psi }_{0\,i,k}^{(0)\,n+1}$ has been obtained, the neutrino–matter stress ${f}_{\nu ,r}$ given by Equation (252) and used in Equations (114) and (121) for the radial hydrodynamics solve is computed as follows. Using Equation (227) and then (229) in Equation (252), we have

$$\begin{eqnarray}\begin{array}{rcl}{f}_{\nu ,r} & = & -\displaystyle \frac{{\rm{\Gamma }}}{\rho w}\displaystyle \frac{4\pi c}{{\left({hc}\right)}^{3}}\,\int {\epsilon }_{0}^{2}d{\epsilon }_{0}\ \displaystyle \sum _{q}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{q}(x,p\right)}_{S}^{(1)}\\ & = & \displaystyle \frac{{\rm{\Gamma }}}{\rho w}\displaystyle \frac{4\pi }{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{3}d{\epsilon }_{0}\ \displaystyle \sum _{q}\displaystyle \frac{{\psi }_{0\,q}^{(1)}({\epsilon }_{0})}{{\lambda }_{q}^{t}({\epsilon }_{0})}\\ & = & -\displaystyle \frac{{\rm{\Gamma }}}{\rho w}\displaystyle \frac{4\pi }{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{3}d{\epsilon }_{0}\ \displaystyle \sum _{q}\displaystyle \frac{1}{3}{{ \mathcal F }}_{q}({\epsilon }_{0})\,{\rm{\Gamma }}{\left(\displaystyle \frac{\partial {\psi }_{0q}^{(0)}({E}_{0})}{\partial R}\right)}_{t,{E}_{0}}\\ & \to & -\displaystyle \frac{1}{\rho }\displaystyle \frac{4\pi }{{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{3}d{\epsilon }_{0}\ \displaystyle \sum _{q}\displaystyle \frac{1}{3}{{ \mathcal F }}_{q}({\epsilon }_{0}){\left(\displaystyle \frac{\partial {\psi }_{0q}^{(0)}({E}_{0})}{\partial R}\right)}_{t,{E}_{0}}\end{array}\end{eqnarray} \tag{ 287 }$$

where in the last equation, we have set Γ and w to their nonrelativistic values of unity. Equation (287) is differenced as

$$\begin{eqnarray}\displaystyle \begin{array}{rcll}{f}_{\nu ,{ri}+\tfrac{1}{2}} & = & -\displaystyle \frac{1}{{\rho }_{i+\tfrac{1}{2}}}\displaystyle \frac{4\pi }{{\left({hc}\right)}^{3}}\displaystyle \frac{1}{3}\sum _{k=1}^{{N}_{k}}\sum _{q=1}^{{N}_{\nu }}{{ \mathcal N }}_{\mathrm{stwt},q}{{ \mathcal F }}_{i+\tfrac{1}{2},k,q}^{n+1}{A}_{i+\tfrac{1}{2}}{\rho }_{i+\tfrac{1}{2}} & \,\displaystyle \frac{{\epsilon }_{0\,i+1,k}^{3}{\rm{\Delta }}{\epsilon }_{0i+1,k}{\psi }_{0\,i+1,k,q}^{(0)}-{\epsilon }_{0\,i,k}^{3}{\rm{\Delta }}{\epsilon }_{0i,k}{\psi }_{0\,i,k,q}^{(0)}}{\tfrac{1}{2}\left({\rm{\Delta }}{M}_{i}+{\rm{\Delta }}{M}_{i+1}\right)}.\end{array}\end{eqnarray} \tag{ 288 }$$

The replacement

$$\begin{eqnarray}\begin{array}{ll}{\epsilon }_{0\,i+\displaystyle \frac{1}{2},k}^{3}{\rm{\Delta }}{\epsilon }_{0i+\displaystyle \frac{1}{2},k}\displaystyle \frac{{\psi }_{0\,i+1,k,q}^{(0)}-{\psi }_{0\,i,k,q}^{(0)}}{\left({R}_{i+1}-{R}_{i}\right)}\to {A}_{i+\tfrac{1}{2}}{\rho }_{i+\tfrac{1}{2}} & \displaystyle \frac{{\epsilon }_{0\,i+1,k}^{3}{\rm{\Delta }}{\epsilon }_{0i+1,k}{\psi }_{0\,i+1,k,q}^{(0)}-{\epsilon }_{0\,i,k}^{3}{\rm{\Delta }}{\epsilon }_{0i,k}{\psi }_{0\,i,k,q}^{(0)}}{\tfrac{1}{2}\left({\rm{\Delta }}{M}_{i}+{\rm{\Delta }}{M}_{i+1}\right)}\end{array}\end{eqnarray} \tag{ 289 }$$

has been made so the final expression for the neutrino stress will have the correct limiting behavior in the short neutrino mean-free-path limit (${{ \mathcal F }}_{i+\tfrac{1}{2},k,q}^{n+1}\to 1$)

$$\begin{eqnarray}&&{f}_{\nu ,{ri}+\tfrac{1}{2}}\to \sum _{k=1}^{{N}_{k}}\sum _{q=1}^{{N}_{\nu }}{{ \mathcal N }}_{\mathrm{stwt},q}\displaystyle \frac{\left({p}_{\nu i+1,k,q}-{p}_{\nu i,k,q}\right){A}_{i+\tfrac{1}{2}}}{\tfrac{1}{2}\left({\rm{\Delta }}{M}_{i}+{\rm{\Delta }}{M}_{i+1}\right)},\end{eqnarray} \tag{ 290 }$$

which is the force per unit mass exerted by an isotropic relativistic gas, and in the long neutrino mean-free-path limit which, with the help of Equation (229), is given by

$$\begin{eqnarray}&&{f}_{\nu ,{ri}+\tfrac{1}{2}}\to \displaystyle \frac{1}{{\rho }_{i+\tfrac{1}{2}}}\displaystyle \frac{4\pi }{{\left({hc}\right)}^{3}}\sum _{k=1}^{{N}_{k}}{\epsilon }_{0\,i,k}^{3}{\rm{\Delta }}{\epsilon }_{0i,k}\sum _{q=1}^{{N}_{\nu }}{{ \mathcal N }}_{\mathrm{stwt},q}\displaystyle \frac{{\psi }_{0\,i+\tfrac{1}{2},k,q}^{(1)}}{{\lambda }_{i+\tfrac{1}{2},k,q}^{t}},\end{eqnarray} \tag{ 291 }$$

where we assume that at large R, where the neutrino mean free paths are large, ${\epsilon }_{0i+1,k}\simeq {\epsilon }_{0i,k}$. As before, ${\rm{\Delta }}{M}_{i}$ denotes the zone mass. Finally, the zone-centered neutrino stress is computed as the average of the edge values, viz.,

$$\begin{eqnarray}&&{f}_{\nu ,{ri}}=\displaystyle \frac{1}{2}\left({f}_{\nu ,{ri}+\tfrac{1}{2}}+{f}_{\nu ,{ri}-\tfrac{1}{2}}\right).\end{eqnarray} \tag{ 292 }$$

To update the lapse given by Equation (262), note first that a comparison of Equation (262) with Equation (252) shows that

$$\begin{eqnarray}&&\left(\displaystyle \frac{4\pi c}{{\rm{\Gamma }}{\left({hc}\right)}^{3}}\int {\epsilon }_{0}^{2}d{\epsilon }_{0}\ \sum _{q}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0q}\right)}_{S}^{(1)}\right)\displaystyle \frac{1}{\rho {{wc}}^{2}}=\displaystyle \frac{{f}_{\nu ,r}}{{{\rm{\Gamma }}}^{2}{c}^{2}}.\end{eqnarray} \tag{ 293 }$$

Setting ${\rm{\Gamma }}=1$, ${M}_{{\rm{g}}}$ [given by Equation (248)] = $M\equiv {M}_{\mathrm{rest}\mathrm{mass}}$, and w = 1, the update of the lapse begins at the outer edge with

$$\begin{eqnarray}&&{a}_{I+1}={a}_{I+\tfrac{1}{2}}=\left(1-\displaystyle \frac{2{{GM}}_{I+\tfrac{1}{2}}}{{c}^{2}{R}_{I+\tfrac{1}{2}}}\right),\end{eqnarray} \tag{ 294 }$$

then inward to the center with

$$\begin{eqnarray}\begin{array}{rcll}{a}_{i} & = & {a}_{i+1} & \exp \left[\left({p}_{i+1}-{p}_{i}-\displaystyle \frac{{\rm{\Delta }}{M}_{i+\tfrac{1}{2}}}{4\pi {R}_{i+\tfrac{1}{2}}^{2}{\rho }_{i+\tfrac{1}{2}}}{f}_{\nu ,{r}{i}+\tfrac{1}{2}}\right)/{\rho }_{i+\tfrac{1}{2}}{c}^{2}\right],\end{array}\end{eqnarray} \tag{ 295 }$$

where the zone-centered quantities defined at zone edges are arithmetic averages of their zone-centered values on either side of the zone edge, and ${\rm{\Delta }}{M}_{i+\tfrac{1}{2}}/4\pi {R}_{i+\tfrac{1}{2}}^{2}{\rho }_{i+\tfrac{1}{2}}$ has been used for ${R}_{i+1}-{R}_{i}$. Zone-edged values of a are defined by ${a}_{i+\tfrac{1}{2}}=({a}_{i}+{a}_{1+1})/2$.

Comparisons of transport tests with a Boltzmann solver revealed a shortcoming of our flux-limited diffusion scheme when encountering a discontinuity in the fluid velocity—namely, a shock. The problem arises from our neglect of the velocity-dependent terms in going from Equation (224) to Equation (225), which should be important for computing the transport at the shock. The reason was traced ostensibly to the fact that both ${\psi }^{(0)}$ and ${\psi }^{(1)}$, being defined with respect to the fluid frame, are therefore physically both discontinuous at a discontinuity in the fluid velocity. However, as a result of the fact ${\psi }^{(1)}$ is given by the gradient of ${\psi }^{(0)}$ (e.g., Equation (229) and its differenced version, Equation (274)) ${\psi }^{(0)}$ is constrained to be continuous across the shock. In the case, in which the fluid velocity ahead of the shock has a large negative radial velocity that discontinuously transitions to a much smaller negative post-shock radial velocity, which characterizes the change in the fluid velocity across the bounce shock as it stagnates and then revives, both ${\psi }^{(0)}$ and ${\psi }^{(1)}$ should exhibit, outwardly, a positive discontinuous change through the shock. However, ${\psi }^{(0)}$, rather than exhibiting this discontinuous behavior, ramps up over many zones to its final pre-shock value ahead of the shock. This behavior of ${\psi }^{(0)}$ is required in order for the required flux through the shock to be computed via Equation (274). This rise of ${\psi }^{(0)}$ through the post-shock region, occurring in the neutrino heating region in the case of the bounce shock, has the unfortunate effect of causing a rise in the mean neutrino energy

$$\begin{eqnarray}&&{\epsilon }_{0\mathrm{rms}}=\sqrt{{\displaystyle \int }_{0}^{\infty }{\psi }_{0}^{(0)}({\epsilon }_{0}){\epsilon }_{0}^{5}d\epsilon /{\displaystyle \int }_{0}^{\infty }{\psi }_{0}^{(0)}({\epsilon }_{0}){\epsilon }_{0}^{3}d\epsilon }.\end{eqnarray} \tag{ 296 }$$

The result is an unphysical additional neutrino heating in the region behind the bounce shock, potentially causing the shock to revive too soon.

We illustrate this problem by comparing 1D simulations performed using Chimera and the relativistic Boltzmann code Agile-Boltztran, which is described in Section 9. The progenitor used is the 15 ${M}_{\odot }$ model evolved to the point of core collapse by Woosley & Heger (2007), and the neutrino physics is that described in Bruenn (1985). The rather straightforward collection of neutrino physics used in this comparison is chosen so that its implementation in both codes can be easily made identical. The unphysical rise in the ${\nu }_{e}$-rms and ${\bar{\nu }}_{e}$-rms energies as computed by Chimera is illustrated by the green lines in Figure 23(a). Compared with the rms energies computed by Agile-Boltztran (plotted by the red lines), the rms energies computed by Chimera are 1–2 MeV higher by the time the neutrinos reach the shock. A consequence of this neutrino rms energy rise is illustrated in Figure 23(b) by the increase in the shock radius due to increased neutrino heating in the Chimera simulation (shown in green) as compared with that given by the Agile-Boltztran simulation (shown in red). The shock position in the Chimera simulations has been pushed about 15 km farther out in radius compared with its position as given by Agile-Boltztran.

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To describe the causes of this problem in more detail and our shock transport algorithm designed to eliminate this problem, consider Equation (274) at a velocity discontinuity situated at zone interface $i+{}^{1}{/}_{2}$. In this circumstance, ${\psi }_{0\,i+1,k}^{(0)\,n+1}$ and ${\psi }_{0\,i,k}^{(0)\,n+1}$ are defined in two different reference frames. In the case of the outwardly directed bounce shock, the former would be defined in the pre-shock frame, the latter in the post-shock frame. To avoid the unphysical ramping up of ${\psi }^{(0)}$ toward the shock in the post-shock frame as we approach the shock from below, ${\psi }_{0\,i,k,q}^{(0)\,n+1}$ is replaced by ${\psi }_{0\,i,k,q}^{(0)\,n+1,\uparrow }$ in Equation (274), where ${\psi }_{0\,i,k,q}^{(0)\,n+1,\uparrow }$ is ${\psi }_{0\,i,k,q}^{(0)\,n+1}$ transformed to the same reference frame as ${\psi }_{0\,i+1,k,q}^{(0)\,n+1}\ ({\epsilon }_{0i});$ i.e., the reference frame of radial zone $i+1$. Note that the energy argument ${\epsilon }_{0i}$, referenced by the subscript k, is the same in the transformed and untransformed function ${\psi }^{(0)}$, reflecting the velocity independence of the Chimera energy grid.

To obtain an expression for ${\psi }_{0\,i,k,q}^{(0)\,n+1,\uparrow }$, let the quantities with subscripts i and i + 1 denote their values defined with respect to the frame corresponding to radial coordinate i and i + 1, respectively. The transformed neutrino functions will be denoted by ${\psi }_{0\,i,k,q}^{(0)\,n+1,\uparrow }$, as above, to emphasize that they are transformed from frame i to frame i + 1 rather than originally residing in radial zone i + 1. Suppressing the indices k, n, and q, we use the invariance of ${f}_{0}({\epsilon }_{0},{\mu }_{0})$—e.g., ${f}_{0i+1}({\epsilon }_{0i+1},{\mu }_{0i+1})={f}_{0i}({\epsilon }_{0i},{\mu }_{0i})$—and the transformation properties of the independent variables ${\epsilon }_{0}$ and ${\mu }_{0}$,

$$\begin{eqnarray}\begin{array}{rclrcl}\beta & \equiv & {\beta }_{i+\tfrac{1}{2}}=\displaystyle \frac{{u}_{i+1}-{u}_{i}}{c},\quad \gamma \equiv {\gamma }_{i+\tfrac{1}{2}}=\displaystyle \frac{1}{\sqrt{1-{\beta }^{2}}}, & {\epsilon }_{0i+1} & = & \gamma {\epsilon }_{0i}(1-\beta {\mu }_{0i}),\quad {\mu }_{0i+1}=\displaystyle \frac{{\mu }_{0i}-\beta }{1-\beta {\mu }_{0i}},\end{array}\end{eqnarray} \tag{ 297 }$$

to derive Equation (298) below, where we have kept terms to order ${ \mathcal O }(v/c)$. The use of ${\psi }_{0\,i,k,q}^{(0)\,n+1,\uparrow }$ in place of ${\psi }_{0\,i,k,q}^{(0)\,n+1}$ in Equation (274) enables the correct flux to be computed through zone interface $i+{}^{1}{/}_{2}$, while the difference between ${\psi }_{0\,i,k,q}^{(0)\,n+1}$ and ${\psi }_{0\,i,k,q}^{(0)\,n+1,\uparrow }$ constitutes the discontinuity in ${\psi }_{0}^{(0)}$ at the shock. It must be noted that the replacement of ${\psi }_{0\,i,k,q}^{(0)\,n+1}$ in Equation (274) by ${\psi }_{0\,i,k,q}^{(0)\,n+1,\uparrow }$ is only performed when the flux is being computed at interface $i+{}^{1}{/}_{2}$. When computed at interface $i-{}^{1}{/}_{2}$, ${\psi }_{0\,i,k,q}^{(0)\,n+1}$ is not modified.

To implement this part of the shock transport algorithm, Chimera computes $d{\psi }_{0\,i,k,q}^{(0)\,\mathrm{shock}}$, the difference between ${\psi }_{0\,i,k,q}^{(0)\,n+1,\uparrow }$ and ${\psi }_{0\,i,k,q}^{(0)\,n+1}$,

$$\begin{eqnarray}\begin{array}{rcl}d{\psi }_{0\,i,k,q}^{(0)\,\mathrm{shock}} & = & {\psi }_{0\,i,k,q}^{(0)\,n+1,\uparrow }-{\psi }_{0\,i,k,q}^{(0)\,n+1}\,={\beta }_{i+\tfrac{1}{2}}\left\{\Space{0ex}{4.4ex}{0ex}-{\psi }_{0\,i,k,q}^{(1)}\right.\left.\,+\displaystyle \frac{{\epsilon }_{0\,i,k+\tfrac{1}{2}}^{3}{\psi }_{0\,i,k+\tfrac{1}{2},q}^{(1)}-{\epsilon }_{0\,i,k-\tfrac{1}{2}}^{3}{\psi }_{0\,i,k-\tfrac{1}{2},q}^{(1)}}{\tfrac{1}{3}\left({\epsilon }_{0\,i,k+\tfrac{1}{2}}^{3}-{\epsilon }_{0\,i,k-\tfrac{1}{2}}^{3}\right)}\right\}\end{array}\end{eqnarray} \tag{ 298 }$$

in the presence of a shock, and $d{\psi }_{0\,i,k,q}^{(0)\,\mathrm{shock}}=0$ otherwise, where

$$\begin{eqnarray}&&{\psi }_{0\,i,k+\displaystyle \frac{1}{2},q}^{(1)}=\displaystyle \frac{1}{2}\left({\psi }_{0\,i,k,q}^{(1)}+{\psi }_{0\,i,k+1,q}^{(1)}\right)\end{eqnarray} \tag{ 299 }$$

and the zone-centered value ${\psi }_{0\,i,k,q}^{(1)}={\eta }_{i,k,q}^{(1)}{\psi }_{0\,i,k,q}^{(0)}$, where η is defined in Section 6.16. With $d{\psi }_{0\,i,k,q}^{(0)\,\mathrm{shock}}$ computed, Equation (274) is modified to read

$$\begin{eqnarray}\begin{array}{rcll}{\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1} & = & -\displaystyle \frac{{\lambda }_{i+\tfrac{1}{2},k,q}^{(t)\,n+1}}{3}{{ \mathcal F }}_{i+\tfrac{1}{2},k,q}^{n+1}{{\rm{\Gamma }}}_{i+\tfrac{1}{2}}^{n+1} & \displaystyle \frac{{\psi }_{0\,i+1,k,q}^{(0)\,n+1}-{\psi }_{0\,i,k,q}^{(0)\,n+1}-d{\psi }_{0\,i,k,q}^{(0)\,n+1,\mathrm{shock}}}{{R}_{i+1}-{R}_{i}}.\end{array}\end{eqnarray} \tag{ 300 }$$

A modification is needed in Equation (273) to complete this scheme. In Equation (273), the change in ${\psi }_{0\,i,k,q}^{(0)\,n+1}$ due to transport is determined by the fluxes through the outer zone edge, $i+{}^{1}{/}_{2}$, proportional to ${\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}$, and by the fluxes through the inner zone edge, $i-{}^{1}{/}_{2}$, proportional to ${\psi }_{0\,i-\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}$. However, ${\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}$ at the outer zone edge is computed in the $i+1$ frame by Equation (300) with ${\psi }_{0\,i,k,q}^{(0)\,n+1\,\uparrow }={\psi }_{0\,i,k,q}^{(0)\,n+1}+d{\psi }_{0\,i,k,q}^{(0)\,n+1,\mathrm{shock}}$ replacing ${\psi }_{0\,i,k,q}^{(0)\,n+1}$, while ${\psi }_{0\,i-\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}$ at the inner zone edge is computed in the ith frame. Consistency is restored by replacing ${\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}$ in Equation (273) by ${\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1,\downarrow }$, where the latter is the former transformed from the frame at $i+1$ to the frame at i. This will ensure that the fluxes computed through both the inner and outer edges of zone i are with respect to the same frame.

To derive an expression for ${\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1,\downarrow }$, we will be transforming quantities from the frame at radial zone $i+1$ to the frame at radial zone i. To keep the notation manageable, we will again suppress the indices k, n, and q, and denote quantities defined at frame i and $i+1$, as before, by subscript i and $i+1$, respectively, and neutrino functions transformed from frame $i+1$ to frame i by a down-arrow to distinguish them from neutrino functions actually residing at radial zone i. In particular, the quantity ${\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}$ is computed in the $i+1$ frame by Equation (300) above, and we will continue to denote its value when transformed to the i frame by ${\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1,\downarrow }$. The transformation is obtained by equating the number of neutrinos passing through an area dS perpendicular to the radial direction in a time dt as seen from the two frames (Mihalas & Mihalas 1984, p.413); namely,

$$\begin{eqnarray}\begin{array}{rcl}{{dN}}_{0i} & = & {\displaystyle \int }_{-1}^{1}d{\mu }_{0i}{f}_{0i}({\epsilon }_{0i},{\mu }_{0i})\,{\epsilon }_{0\,i}^{2}\,d{\epsilon }_{0i}\,c\,{\mu }_{0i}\,{{dSdt}}_{0i},\mathrm{and}\\ & = & {{dN}}_{0i+1}={\displaystyle \int }_{-1}^{1}d{\mu }_{0i+1}{f}_{0i+1}({\epsilon }_{0i+1},{\mu }_{0i+1})\,{\epsilon }_{0\,i+1}^{2}\,d{\epsilon }_{0i+1}\,c\,{\mu }_{0i+1}\,{{dSdt}}_{0i+1}.\end{array}\end{eqnarray} \tag{ 301 }$$

Using ${\epsilon }_{0i+1}\,{\mu }_{0i+1}\,d{\mu }_{0i+1}={\epsilon }_{0i}\,{\mu }_{0i}\,d{\mu }_{0i}$ in the expression for ${{dN}}_{0i+1}$, and transforming the remaining variables to the i-frame, we get

$$\begin{eqnarray}\begin{array}{rcl}{{dN}}_{0i+1} & = & {\displaystyle \int }_{-1}^{1}d{\mu }_{0i}{f}_{0i+1}\left(\gamma {\epsilon }_{0i}(1-\beta {\mu }_{0i}),\displaystyle \frac{{\mu }_{0i}-\beta }{1-\beta {\mu }_{0i}}\right)\\ & & \times \gamma {\epsilon }_{0i}(1-\beta {\mu }_{0i}){\epsilon }_{0i}\,d{\epsilon }_{0i}\,c\,\displaystyle \frac{{\mu }_{0i}-\beta }{1-\beta {\mu }_{0i}}\,{dS}\,\gamma \,{{dt}}_{0i}.\end{array}\end{eqnarray} \tag{ 302 }$$

Dividing by $c\,{\epsilon }_{0\,i}^{2}\,d{\epsilon }_{0i}\,{dS}\,{{dt}}_{0i}$ and considering terms to order ${ \mathcal O }(v/c)$,

$$\begin{eqnarray}\begin{array}{rcl}{\psi }_{0\,i+1}^{(1)\,\downarrow }({\epsilon }_{0i}) & = & {\displaystyle \int }_{-1}^{1}d{\mu }_{0i}{f}_{0i}({\epsilon }_{0i},{\mu }_{0i})\,{\mu }_{0i}\,=\,\displaystyle \frac{{{dN}}_{0i}}{c\,{\epsilon }_{0\,i}^{2}\,d{\epsilon }_{0i}\,{dS}\,{{dt}}_{0i}}=\displaystyle \frac{{{dN}}_{0i+1}}{c\,{\epsilon }_{0\,i}^{2}\,d{\epsilon }_{0i}\,{dS}\,{{dt}}_{0i}}\\ & & \simeq {\displaystyle \int }_{-1}^{1}d{\mu }_{0i}{f}_{0i+1}\left({\epsilon }_{0i}(1-\beta {\mu }_{0i}),({\mu }_{0i}-\beta )\right.\left.(1+\beta {\mu }_{0i}\right)](1-\beta {\mu }_{0i})\,({\mu }_{0i}-\beta )(1+\beta {\mu }_{0i})\\ & \simeq & {\displaystyle \int }_{-1}^{1}d{\mu }_{0i}{f}_{0i+1}[\left({\epsilon }_{0i}-\beta {\epsilon }_{0i}{\mu }_{0i}\right),\left({\mu }_{0i}-\beta \right.)\left.(1-{\mu }_{0\,i}^{2},)\right)\,({\mu }_{0i}-\beta )\\ & \simeq & {\displaystyle \int }_{-1}^{1}d{\mu }_{0i}\left[{\mu }_{0i}{f}_{0i+1}\left({\epsilon }_{0i},{\mu }_{0i}\right)-{\mu }_{0i}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial {\mu }_{0}}\right)}_{i}\right.\beta (1-{\mu }_{0\,i}^{2})-{\mu }_{0i}{\left(\displaystyle \frac{\partial {f}_{0}}{\partial {\epsilon }_{0}}\right)}_{i}\beta {\epsilon }_{0i}{\mu }_{0i}-\left.\,\beta {f}_{0i+1}\left({\mu }_{0i},{\epsilon }_{0i}\right)\Space{0ex}{3.70ex}{0ex}\right]\\ & = & {\psi }_{0\,i+1}^{(1)}({\epsilon }_{0i})-\displaystyle \frac{\beta }{{\epsilon }_{0\,i}^{2}}\left(\displaystyle \frac{\partial {\epsilon }_{0\,i}^{3}{\psi }_{0\,i+1}^{(2)}({\epsilon }_{0i})}{\partial {\epsilon }_{0i}}\right).\end{array}\end{eqnarray} \tag{ 303 }$$

To implement this part of the shock transport algorithm, Chimera computes $d{\psi }_{0\,i+\tfrac{1}{2},q}^{(1)\,\mathrm{shock}}({\epsilon }_{0i})$, the difference between ${\psi }_{0\,i+1,q}^{(1)\,\downarrow }\,({\epsilon }_{0i})$ and ${\psi }_{0\,i+1,q}^{(1)}({\epsilon }_{0i})$, both defined at zone edge $i+{}^{1}{/}_{2}$, by

$$\begin{eqnarray}&&\begin{array}{l}d{\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,\mathrm{shock}}\,=\,-{\beta }_{i+\tfrac{1}{2}}\displaystyle \frac{{\epsilon }_{0\,i,k+\tfrac{1}{2}}^{3}{\psi }_{0\,i+\tfrac{1}{2},k+\tfrac{1}{2},q}^{(2)}-{\epsilon }_{0\,i,k-\tfrac{1}{2}}^{3}{\psi }_{0\,i+\tfrac{1}{2},k-\tfrac{1}{2},q}^{(2)}}{\tfrac{1}{3}\left({\epsilon }_{0\,i,k+\tfrac{1}{2}}^{3}-{\epsilon }_{0\,i,k-\tfrac{1}{2}}^{3}\right)},\end{array}\end{eqnarray} \tag{ 304 }$$

in the presence of a shock, and by $d{\psi }_{0\,i+\tfrac{1}{2},k,q}^{(1)\,\mathrm{shock}}=0$, otherwise, where

$$\begin{eqnarray}&&{\psi }_{0\,i,k+\displaystyle \frac{1}{2},q}^{(2)}=\displaystyle \frac{1}{2}\left({\psi }_{0\,i,k,q}^{(2)}+{\psi }_{0\,i,k+1,q}^{(2)}\right)\end{eqnarray} \tag{ 305 }$$

and where ${\psi }_{0\,i,k,q}^{(2)}={\eta }_{i,k,q}^{(2)}{\psi }_{0\,i,k,q}^{(0)}$. With $d{\psi }_{0\,i+\tfrac{1}{2},k,q}^{(1)\,\mathrm{shock}}$ computed, Equation (273) is modified to read

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{{a}_{i}c}\displaystyle \frac{{\psi }_{0\,i,k,q}^{(0)\,n+1}-{\psi }_{0\,i,k,q}^{(0)\,n}}{{\rm{\Delta }}{t}^{n+\tfrac{1}{2}}}\\ \qquad +\,\displaystyle \frac{{a}_{i}^{2}}{{\mathrm{Vol}}_{i}}\left[\displaystyle \frac{1}{{a}_{i+\tfrac{1}{2}}^{2}}{\mathrm{Area}}_{i+\displaystyle \frac{1}{2}}\left({\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}+d{\psi }_{0\,i+\displaystyle \frac{1}{2},k,q}^{(1)\,\mathrm{shock}}\right)\right.\left.\,-\,\displaystyle \frac{1}{{a}_{i-\tfrac{1}{2}}^{2}}{\mathrm{Area}}_{i-\displaystyle \frac{1}{2}}{\psi }_{0\,i-\displaystyle \frac{1}{2},k,q}^{(1)\,n+1}\right] & =\,\displaystyle \frac{c}{{\epsilon }_{0}}{\left(\displaystyle \frac{d}{d{\ell }}{f}_{0}(x,p\right)}_{S\ i,k,q}^{(0)\,n+1}.\end{array}\end{eqnarray} \tag{ 306 }$$

Numerical experiments have shown that rolling in this algorithm from $| {v}_{i+1}-{v}_{i}|$ = 0.01–0.02 c with a corresponding rollout of the energy advection algorithm—i.e., at the shock, this algorithm replaces the energy-advection algorithm—gave excellent results. In the above comparisons of the neutrino rms energies and shock trajectories computed with the Chimera and Agile-Boltztrancodes, the Chimera results with the use of the above shock transport algorithm, plotted in Figure 23 by the lines in blue, show much better agreement with the results from Agile-Boltztran. The discontinuities in ${\psi }_{0}^{(0)}$ are clearly evident, as is the absence of the unphysical rise in ${\psi }_{0}^{(0)}$ as the shock is approached from below. Further examples of Chimera test results without the use of this scheme (Series B) and with the use of this scheme (Series C) are given in Sections 7 and 9 and demonstrate the scheme's ability to give accurate transport solutions across a shock.

As a test of the ability of Chimera to conserve total energy, the simulations described above were redone with the Newtonian gravitational potential substituted for the general relativistic monopole potential. Substitution of the general relativistic monopole in place of the Newtonian monopole in the multipole expansion of the self-gravitational potential is not conservative and, therefore, not suitable for a test of total energy conservation. We also set the lapse function to unity. The results are shown in Figure 23, Panels (c) and (d). In both simulations, energy is conserved with high accuracy up to bounce, where a discontinuous change in energy of about −0.35 B occurs in both simulations. Thereafter, the total energy of the simulation with the shock transport algorithm turned off is essentially flat for the next ∼140 ms, after which it decreases by about −0.1 B during the subsequent ∼100 ms. The total energy of the simulation with the shock transport algorithm turned on increases by about 0.1 B for the ∼140 ms following bounce and then remains essentially flat for the next ∼100 ms. The components of the total energy of the two Chimera simulations are plotted in Panel (d). There are small differences discernible in the components of the total energy in the two simulations, particularly between 300 and 400 ms.

Following the Lagrangian transport step, Chimera executes the Lagrangian energy advection step, which consists of solving Equation (241), which has been operator split along with the "transport" Equation (240) from Equation (222). Omitting the superscript E from ${\left(\partial {\psi }_{0}^{(0)}/\partial t\right)}_{m,{E}_{0}}$ with the understanding that this time derivative will hereafter refer to the change in ${\psi }_{0}^{(0)}$ due to the energy advection step, and considering for the moment only the terms in Equation (241) involving ${\psi }_{0}^{(0)}$ (terms involving ${\psi }_{0}^{(2)}$ will be considered shortly), we rearrange these terms as follows: