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≤ γ h 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 γ is independent of h and τ. However, error estimates forhigher order Pᵏ(k≥ 2) elements need a more restrictive time step τ≤ γ h4/3. We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition τ≤γ h for the Pᵏ elements of degree k≥ 2. Error estimates of O(hk+1/2+τ²) are obtained for general monotone numerical fluxes, and optimal error estimates of O(hᵏ⁺¹+τ²) are obtained for upwind numerical fluxes.