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 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, and time is advanced by the third order explicit total variation diminishing Runge-Kutta method under the standard CFL temporal-spatial condition. The error at the final time T in the L(R\RT )-norm is the optimal order both in space and in time, where RT is the pollution region due to the initial discontinuity with the width of order O(h1/2 log(1/h)), where h is the maximum cell length.