Semi-implicit spectral deferred correction methods for highly nonlinear partial differential equations

Abstract

The goal of this paper is to develop a novel semi-implicit spectral deferred correction (SDC) time marching method. The method can be used in a large class of problems, especially for highly nonlinear ordinary differential equations (ODEs) without easily separating of stiff and non-stiff components, which is more general and efficient comparing with traditional semi-implicit SDC methods. The proposed semi-implicit SDC method is based on low order time integration methods and corrected iteratively. The order of accuracy is increased for each additional iteration. And we also explore its local truncation error analytically. This SDC method is intended to be combined with the method of lines, which provides a flexible framework to develop high order semi-implicit time marching methods for nonlinear partial differential equations (PDEs). In this paper we mainly focus on the applications of the nonlinear PDEs with higher order spatial derivatives, e.g. convection diffusion equation, the surface diffusion and Willmore flow of graphs, the Cahn–Hilliard equation, the Cahn–Hilliard–Brinkman system and the phase field crystal equation. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the fully discrete schemes are all high order accurate in both space and time, and stable numerically with the time step proportional to the spatial mesh size. Numerical experiments are carried out to illustrate the accuracy and capability of the proposed semi-implicit SDC method.