Error estimates for the third order explicit Runge-Kutta discontinuous Galerkin method for a linear hyperbolic equation in one-dimension with discontinuous initial data

In this paper we present an error estimate for the explicit Runge-Kutta discontinuous Galerkin method to solve a linear hyperbolic equation in one dimension with discontinuous but piecewise smooth initial data. The discontinuous finite element space is made up of piecewise polynomials of arbitrary degree $$k\ge 1$$k≥1, and time is advanced by the third order explicit total variation diminishing Runge-Kutta method under the standard CFL temporal-spatial condition. The $$L^2(\mathbb R \backslash \mathcal R _T)$$L2(R∖RT)-norm error at the final time $$T$$T is optimal in both space and time, where $$\mathcal R _T$$RT is the pollution region due to the initial discontinuity with the width $$\mathcal O (\sqrt{T\beta }h^{1/2}\log (1/h))$$O(Tβh1/2log(1/h)). Here $$h$$h is the maximum cell length and $$\beta $$β is the flowing speed. These results are independent of the time step and hold also for the semi-discrete discontinuous Galerkin method.