Optimal Error Estimates of the Semi-Discrete Local Discontinuous Galerkin Method and Exponential Time Differencing Schemes for the Thin Film Epitaxy Problem Without Slope Selection

Abstract

In this paper, we prove the optimal error estimates in $L^2$ norm of the semi-discrete local discontinuous Galerkin (LDG) method for the thin film epitaxy problem without slope selection. To relax the severe time step restriction of explicit time marching methods, we employ a class of exponential time differencing (ETD) schemes for time integration, which is based on a linear convex splitting principle. Numerical experiments of the accuracy and long time simulations are given to show the efficiency and capability of the proposed numerical schemes.