Stability analysis and a priori error estimates of the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws

摘要

In this paper we present an analysis of the Runge-Kutta discontinuous Galerkin method for solving scalar conservation laws, where the time discretization is the third order explicit total variation diminishing Runge-Kutta method. We use an energy technique to prove the L~2-norm stability for scalar linear conservation laws and to obtain a priori error estimates for smooth solutions of scalar nonlinear conservation laws. Quasi-optimal order is obtained for general numerical fluxes, and optimal order is given for upwind fluxes. The theoretical results are obtained for piecewise polynomials with any degree k ≥ 1 under the standard temporal-spatial CFL condition τ ≤ γh, where h and τ are the element length and time step, respectively, and the positive constant γ is independent of h and τ.