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 a0Uxx on the right hand side of the partial differential equation, then to treat the term a0Uxx implicitly and the other terms (a(U)Ux)x − a0Uxx 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 a0, i.e, a0 ≥ max{a(u)}/2 and 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.