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

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 Ut= (a(U)Ux)x. The basic idea is to add and subtract two equal terms a0Uxxon the right-hand side of the partial differential equation, then to treat the term a0Uxximplicitly and the other terms (a(U)Ux)x− a0Uxxexplicitly. 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 a0, i.e., a0≽ 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.