Superconvergence of Discontinuous Galerkin Methods for Linear Hyperbolic Equations

Abstract

In this paper, we study superconvergence properties of the discontinuous Galerkin (DG) method for one-dimensional linear hyperbolic equations when upwind fluxes are used. We prove, for any polynomial degree $k$, the $2k+1$th (or $2k+1/2$th) superconvergence rate of the DG approximation at the downwind points and for the domain average under quasi-uniform meshes and some suitable initial discretization. Moreover, we prove that the derivative approximation of the DG solution is superconvergent with a rate $k+1$ at all interior left Radau points. All theoretical findings are confirmed by numerical experiments.