Optimal Error Estimates of the Local Discontinuous Galerkin Method and High-order Time Discretization Scheme for the Swift–Hohenberg Equation

In this paper, we develop a local discontinuous Galerkin (LDG) method for the Swift–Hohenberg equation. The energy stability and optimal error estimates in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} norm of the semi-discrete LDG scheme are established. To avoid the severe time step restriction of explicit time marching methods, a first-order linear scheme based on the scalar auxiliary variable (SAV) method is employed for temporal discretization. Coupled with the LDG spatial discretization, we achieve a fully-discrete LDG method and prove its energy stability and optimal error estimates. To improve the temporal accuracy, the semi-implicit spectral deferred correction (SDC) method is adapted iteratively. Combining with the SAV method, the SDC method can be linear, high-order accurate and energy stable in our numerical tests. Numerical experiments are presented to verify the theoretical results and to show the efficiency of the proposed methods.