Numerical simulation of the coupled viscous Burgers equation using the Hermite formula and cubic B-spline basis functions

A numerical procedure dependent on the cubic B-spline and the Hermite formula is developed for the coupled viscous Burgers’ equation (CVBE). The method uses a combination of the Hermite formula and the cubic B-spline for discretization of the space dimension while the time dimension is approximated using the typical finite differences. A piecewise continuous sufficiently smooth function is obtained as a solution which allows to approximate solution at any location in the domain of interest. The scheme is tested for stability analysis and is proved to be unconditionally stable. Numerical experiments and comparison of outcomes reveal that the suggested scheme comes up with better accuracy and is extremely productive.

The CVBE was first used by Esipov [1] to study the model of polydispersive sedimentation. Numerous authors have investigated the coupled linear and nonlinear initial/ boundary value problems. Nee and Duan [2] studied a coupled system of Burgers' equations with zero Dirichlet boundary conditions and appropriate initial data. The harmonic differential quadrature finite differences coupled methodology [2] and conjugate channel approach [3] is available to deal with nonlinear coupled equations. A mesh free numerical procedure was proposed in [4] for the coupled non-linear PDEs. Khater et al [5] acquired the numerical solution of the VCBE utilizing the cubic-spline collocation method. Deghan et al [6] used Adomian Pade procedure to Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. deal with the coupled Burgers' equation (CBE). Rashid and Ismail et al [7] used Fourier Pseudospectral technique to discover the numerical solution of the CVBE. Adbou and Soliman [8] used variational iteration method for solving the CVBE. An explicit solution of the CVBE was obtained by Kaya [9] using Adomian decomposition method. Soliman [10] presented a modified extended tanh-function method for Burger like equations. Abazari and Borhanifar [11] obtained the numerical solution of the Burgers' and CBE using the differential transformation method. Further numerical solution of the CVBE using cubic B-spline functions is obtained by Mittal and Arora [12]. Mokhtari et al [13] presented generalized differential quadrature method for the Burgers' equation. Srivastava et al [14][15][16] proposed various finite difference methods for the two dimensional CVBE. Srivastava et al [17] proposed an implicit logarithmic finite difference method for the one-dimensional CBE. Higher order trigonometric B-spline based algorithms were presented in [18] to numerically study the coupled Burgers' equation.
Stimulated by the success of the spline approach in finding numerical solutions of partial differential equations, we have used combination of the Hermite formula and cubic B-spline for approximating the space derivative. This merger has considerably augmented the accuracy of the scheme. Another advantage is that the approximate solution is come up as a smooth piecewise continuous function allowing one to get solution at any wanted location in the domain.
The rest of the paper is assembled as follows. In section 2, the numerical scheme is derived thoroughly. In section 3, the stability of the scheme is talked about. Section 4 offers a contrast of our numerical consequences with the ones presented earlier on. Section 5 sums up the outcomes of this work.

The derivation of the scheme
. The approximate solutions V(z, t) and W(z, t) to the exact solutions v(z, t) and w(z, t) of (1.1) are found as are survived at the grid point, z j on account of the local support property of CuBS. Accordingly, the approximations v j n and w j n at n th time level are given by   The time derivative is discretized by finite difference scheme and space derivative by the Crank-Nicolson scheme so that (1.1) takes the form  Note that the nonlinear advection terms vv z , ww z , vw z and wv z are usually discretized as: Using (2.7) in (2.5) and (2.6), we obtain and The Hermite Formula [19] is given by Using (2.10) in (2.8) and (2.9), we obtain and          1 . Four additional equations are needed to obtain a consistent system. These equations can be extracted from the given boundary conditions. Now the resulting system can be uniquely solved using any Gaussian elimination based algorithm.

Initial State:
To start iterations, the initial vector The system produces an ( ) ( ) +´+ N N 2 6 2 6 matrix system. The required initial vector is unique solution of this system.    -N=200 N=400 N=200 N=400 N=200 N=400

Stability analysis
In this section, the stability of the proposed scheme (2.6) is proved which shows that the scheme is unconditionally stable for whole of the domain. For this purpose, we first change vv z , ww z , vw z and wv z to a linear terms by substituting v and w as a constant d 1 and d 2 as is done in Von Neumann method. The linearized form of (2.5) is given as  which on simplification by utilizing Hermite Formula [19] reduces to         Now substituting the Fourier modes , where A and B are harmonics amplitudes, ξ is the mode number, h is the element size and i = -1 in (3.3), we obtain   Since [ ] f p p Î -, , without loss of generality, we can assume that f=0 so that (3.6) becomes which proves (2.5) is unconditionally stable. Since (2.5) and (2.6) are symmetric in v and w, similar results can be obtained from (2.6).

Numerical experiments and discussion
In this section the accuracy of the proposed scheme is verified by some test problems and is measured with two discrete L 2 and ¥ L error norms defined as , .
n j j n j n and the order of convergence is given by [12] ( ( ) The scheme is applied to this problem to check its accuracy. In figure 1 the numerical and exact solutions at different times are compared with tremendous closeness. Figure 2 exhibits the absolute errors for v and w when N=50, Δt=0.001 and t=1. Figures 3 and 4 plot the 3D contrast between the exact and approximate solutions. Tables 1 and 2 compare computed errors with the ones computed in [7,12,17]. The approximate solutions V(z, t) and W(z, t) when N=20, Δt=0.01 and t=1 for example 1 are  given by ) ) ) ) ) ) ( )  ) ) ) ( ) We utilize the proposed scheme to acquire numerical results. Figure 5 plots the error profiles when Δt=0.01, N=100 and t=1 for v and w. In figures 6 and 7 the 3D comparisons are shown between the exact and numerical solutions. An excellent comparison of error norms is tabulated in tables 3 and 4.
The approximate solutions V(z, t) and W(z, t) when N=20, Δt=0.01 and t=1 for example 2 are given by and Numerical calculations are performed using Δt=0.001 and N=50. The maximum values of v and w are computed and are compared with those presented in [12,17]. The results are tabulated in tables 5 and 6. Table 7 compares absolute errors with the ones listed in [12]. In figure 8, a comparison between the exact and approximate solutions is shown for various values of α, β, η and ζ. Figures 9 and 10 plots 3D approximate and exact solutions for various values of α and β. The approximate solutions V(z, t) and W(z, t) when N=20, α=β=10, η=ζ=2, Δt=0.001 and t=0.1 for example 3 are given by

Concluding remarks
This investigation presents a numerical procedure dependent on cubic B-spline and the Hermite formula for the CVBE. This technique uses the standard finite differences to discretize the time dimension while the space dimension is approximated using the Hermite formula and the cubic B-spline. The refinement of the scheme using the Hermite formula has appreciably increased the accuracy of the scheme. The stability of the scheme has been checked to affirm that it is unconditionally stable. Numerical and graphical comparisons reveal that the presented procedure is computationally better and effective. It is worthy of mention that the scheme can be tried to a variety of partial differential equations.