Error Estimates for the Runge–Kutta Discontinuous Galerkin Method for the Transport Equation with Discontinuous Initial Data

Abstract

We study the approximation of nonsmooth solutions of the transport equation in one space dimension by approximations given by a Runge–Kutta discontinuous Galerkin method of order two. We take an initial datum, which has compact support and is smooth except at a discontinuity, and show that, if the ratio of the time step size to the grid size is less than $1/3$, the error at the time T in the $L^2(\mathbb{R}\setminus\mathcal{R}_T)$-norm is the optimal order two when $\mathcal{R}_T$ is a region of size $O(T^{1/2}\,h^{1/2}\;\log{1/h})$ to the right of the discontinuity and of size $O(T^{1/3}\,h^{2/3}\;\log{1/h})$ to the left. Numerical experiments validating these results are presented.