ERROR ESTIMATES TO SMOOTH SOLUTIONS OF RUNGE--KUTTA DISCONTINUOUS GALERKIN METHODS FOR SCALAR CONSERVATION LAWS.

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 P¹ (piecewise linear) elements are obtained under the usual CFL condition τ ≤ γh for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions, where h and τ are the maximum element lengths and time steps, respectively, and the positive constant yγ is independent of h and τ. However, error estimates for higher order Pk (k ≥ 2) elements need a more restrictive time step τ ≤ γh4/3. We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition τ ≤ γh for the Pk elements of degree k ≥ 2. Error estimates of O(hk+½ + τ²) are obtained for general monotone numerical fluxes, and optimal error estimates of O(hk+1 + τ²) are obtained for upwind numerical fluxes. [ABSTRACT FROM AUTHOR]