A Priori Error Estimates To Runge-Kutta Discontinuous Galerkin Finite Element Method For Symmetrizable Systems Of Conservation Laws With Sufficiently Smooth Solution

In this paper we present an analysis of the Runge-Kutta discontinuous Galerkin method for solving symmetrizable conservation laws, where the time discretization is the third order explicit total variation diminishing Runge-Kutta method and the finite element space is kth piecewise polynomial space Pk. We use energy technique to prove the stability for symmetrizable conservation laws and to obtain a priori error estimate for smooth solutions for nonlinear conservation laws. Quasi-optimal order is obtained for general monotone numerical fluxes under the periodic boundary condition. The theoretical results are obtained for piecewise polynomials with degree κ≥ 2 under the standard temporal-spatial CFL condition τ≤γh, where h is the element length and τ is time step, and γ is a positive constant independent of h and τ.