Error Estimate Of The Local Discontinuous Galerkin Method For Nonlinear Convection-diffusion Equation

In nature, nonlinear convection-diffusion equation can almost describe many natural phenomena and practical problems, such as water pollution, heat transmission and oil recovery simulation, and this kind of equation generally cannot get the exact solution.Therefore, research on numerical solution and error estimate has practical application value. In this paper, the main research is to use local discontinuous Galerkin method（LDG） to analysis the error estimate in one dimension and the multidimensional Cartesian meshes, to define the numerical flux and projection, and to give the cell entropy inequality and2L-analysis.In this paper, we first rewrite the nonlinear convection-diffusion equation into a system with first-order equations, and then to apply the discontinuous Galerkin method（DG） to the first-order system to get the discrete format. We also discuss the value of the numerical flux. In order to ensure the error does not increase with the solution process, namely the solution can be controlled by the initial value, we analyse the cell entropy inequality to prove the stability in one dimension and multidimensional Cartesian meshes.Further, the most important research of this paper is error estimate. In one dimension, we generally use the energy equation to estimate the error of KdV equation,however, in this paper, we use the energy norm and Taylor expansion to analyze it. By choosing appropriate projection, we can eliminate many boundary items, and get the error estimate. The idea has also been applied to the case of multidimensional Cartesian meshes, the only difference is the definition and property of the projection.Finally, by the above method, we have proven the error estimate of order O (h^k+1/2)in one dimension and multidimensional Cartesian meshes.