Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws

摘要

In this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for Runge-Kutta discontinuous Galerkin (RKDG) methods, where the time discretization is the second order explicit total variation diminishing ( TVD) Runge-Kutta method. Error estimates for the P-1 (piecewise linear) elements are obtained under the usual CFL condition tau less than or equal to gammah for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions, where h and tau are the maximum element lengths and time steps, respectively, and the positive constant gamma is independent of h and tau. However, error estimates for higher order P-k(k greater than or equal to 2) elements need a more restrictive time step tau less than or equal to gammah(4/3). We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition tau less than or equal to gammah for the P-k elements of degree k greater than or equal to 2. Error estimates of O(h(k+1/2) + tau(2)) are obtained for general monotone numerical fluxes, and optimal error estimates of O(h(k+1) + tau(2)) are obtained for upwind numerical fluxes.