Stability analysis and error estimates of Lax–Wendroff discontinuous Galerkin methods for linear conservation laws^∗

Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. ;
zhengsun@brown.edu

Received: 30 March 2016
Revised: 21 June 2016
Accepted: 5 July 2016

Abstract

In this paper, we analyze the Lax–Wendroff discontinuous Galerkin (LWDG) method for solving linear conservation laws. The method was originally proposed by Guo et al. in [W. Guo, J.-M. Qiu and J. Qiu, J. Sci. Comput. 65 (2015) 299–326], where they applied local discontinuous Galerkin (LDG) techniques to approximate high order spatial derivatives in the Lax–Wendroff time discretization. We show that, under the standard CFL condition τ ≤ λh (where τ and h are the time step and the maximum element length respectively and λ> 0 is a constant) and uniform or non-increasing time steps, the second order schemes with piecewise linear elements and the third order schemes with arbitrary piecewise polynomial elements are stable in the L² norm. The specific type of stability may differ with different choices of numerical fluxes. Our stability analysis includes multidimensional problems with divergence-free coefficients. Besides solving the equation itself, the LWDG method also gives approximations to its time derivative simultaneously. We obtain optimal error estimates for both the solution u and its first order time derivative u_t in one dimension, and numerical examples are given to validate our analysis.