Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

Abstract

In this paper we discuss the local discontinuous Galerkin methods coupled with two specific explicit-implicit-null time discretizations for solving one-dimensional nonlinear diffusion problems $U_t=(a(U)U_x)_x$. The basic idea is to add and subtract two equal terms $a_0 U_{xx}$ on the right hand side of the partial differential equation, then to treat the term $a_0 U_{xx}$ implicitly and the other terms $(a(U)U_x)_x-a_0 U_{xx}$ explicitly. We give stability analysis for the method on a simplified model by the aid of energy analysis, which gives a guidance for the choice of $a_0$, i.e, $a_0 \ge \max\{a(u)\}/2$ to ensure the unconditional stability of the first order and second order schemes. The optimal error estimate is also derived for the simplified model, and numerical experiments are given to demonstrate the stability, accuracy and performance of the schemes for nonlinear diffusion equations.