Stability Analysis and Error Estimation of Variable Coefficient Convection-Diffusion Equation: Generalized Numerical Fluxes

In this paper, we study the discontinuous Galerkin (DG) method with generalized numerical fluxes for one-dimensional variable coefficient convection-diffusion equation. For the convection term, we choose the upwind numerical flux, and for the diffusion term, we choose a special type of generalized numerical fluxes, thus we first show that the L 2 stability of the DG scheme. Then, by introducing the projection method, we are able to show k order optimal error estimates for the DG scheme. Finally, Numerical experiment is provided to verify the theoretical results.

1  k order optimal error estimate. In 2017, Cao [4], Li, Yang, and Zhang studied the superconvergence properties of the DG method with upwind-biased flux for linear hyperbolic laws, and obtained 1 2  k order superconvergence. In 2019, Li [5], Zhang, Meng, and Wu gave a DG scheme for selecting the upwind numerical flux for linear hyperbolic laws with degenerate variable coefficients, the stability of the scheme and the 1  k order optimal error estimate are obtained. In the same year, Fu [6], Cheng, Li and Xu analyzed the DG method for several partial differential equations with higher order spatial derivatives. Identify a sub-family of numerical flux by selecting the coefficients in the linear combination, so that the solution of the proposed DG method and some auxiliary variables have the best accuracy under the norm. In 2020, Li [7], Zhang, Meng, Wu and Zhang studied the DG methods 1  k order error estimate of the DG scheme . In this paper, we concentrate on DG method with generalized fluxes for one-dimensional variable coefficient convection-diffusion equation.
 are assumed to be sufficiently smooth with respect to x for simplicity, the article assumes     0 ,and periodic boundary conditions are considered.
The organization of this paper is as follows. In Section 2, the DG scheme of the variable coefficient convection-diffusion equation under the a special type of generalized numerical fluxes and the stability analysis is presented. In Section 3, by introducing a special global projection, the k order optimal error estimate is obtained. In Section 4, numerical experiment is shown, which confirms the k order of optimal error estimates. Concluding remarks are given in section 5.

Basic notation.
In this paper, we use the following usual notation of the DG methods. Denote computational interval as   Since the complexity of the equations expression in the proof, thus we introduce the DG discrete operator, V v  , there holds, for more details, see [8].

The DG scheme
Refering to the variable coefficient convection-diffusion equation (1), the DG scheme is as follows: are numerical fluxes. Since the selection of the convection term and the diffusion term is independent of each other, the selection of the numerical fluxes in this paper is as follows, for the convection term, we choose Let's verify that the semi-discrete discontinuous finite element scheme has 2 L stability by selecting the above two numerical fluxes. Theorem 1. ( 2 L stability) The DG scheme(2) is 2 L stable under numerical fluxes (3) and (4), Proof. From the choice of numerical fluxes and the definition of DG discrete operator, the following formula is established (6) and summing up over all j , then equation (6) can be written as By the periodic boundary conditions, we have by the Newton Leibniz formula, we have Then, by the Cauchy Schwartz's inequality, we get Thus, equation (7) becomes By the inverse inequality(iii), it is easy to show that by Young inequality, after some simple algebraic calculations The 2 L stability result can be obtained by integrating the above inequality with respect to time between 0 andT . This ends the proof.

Optimal error estimates.
This section is devoted to proof of optimal error estimates of DG methods with (2) for variable coefficient convection-diffusion equation.
Let's first introduce the following projection . For , then there holds the following optimal approximation property: where C is independent of h , for more details, see [9] [10]. As for the initial discretization, we usually use the following standard where C is independent of h .
Proof. We will finish the proof with the following two steps.
Step 1: Error equation. Since the exact solution u satisfies the DG scheme(2) and by Galerkin orthogonality, there holds the the following error equation at each element,  (20) and summing up over all j , the error equation (20) can be written Step 2:Estimate of  ， terms. Let ，combining with the conclusions in the stability analysis process to Then, equation (22)

Numerical experiments.
In this section, we provide a numerical experiment to validate the theoretical results. For the example, the third order TVD Runge-Kutta time discretization is used with a suitable time step.
Example 4.1 Consider the following variable coefficient convection-diffusion equation: