Stability analysis and error estimates of Lax-Wendroff discontinuous Galerkin methods for linear conservation laws

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 [11], 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 L 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 ut in one dimension, and numerical examples are given to validate our analysis.