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 methodscoupled with two specific explicit-implicit-null time discretizations forsolving 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 themethod 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, andnumerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations.