Local Analysis of Local Discontinuous Galerkin Method for the Time-Dependent Singularly Perturbed Problem

Abstract

In this paper we will present the local stability analysis and local error estimate for the local discontinuous Galerkin (LDG) method, when solving the time-dependent singularly perturbed problems in one dimensional space with a stationary outflow boundary layer. Based on a general framework on the local stability, we obtain the optimal error estimate out of the local subdomain, which is nearby the outflow boundary point and has the width of $$\mathcal {O}(h\log (1/h))$$O(hlog(1/h)), for the semi-discrete LDG scheme and the fully-discrete LDG scheme with the second order explicit Runge---Kutta time-marching. Here $$h$$h is the maximum mesh length. The numerical experiments are given also.