Abstract
In this paper we shall establish an a priori L$^2$-norm error estimate of the fourth order Runge--Kutta discontinuous Galerkin method for solving sufficiently smooth solutions of one-dimensional scalar nonlinear conservation laws. The optimal order of accuracy in time is obtained under the standard Courant--Friedrichs--Lewy condition, and the quasi-optimal and/or optimal order of accuracy in space is achieved for many widely used numerical fluxes, no matter whether the solution contains sonic points or not. The main tools used in this paper are the matrix transferring process and the generalized Gauss--Radau projection of the reference functions, depending on the relative upwind effect. Finally some numerical experiments are given to support our theoretical results.
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