Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations

Abstract

In this paper, we study the superconvergence properties of the energy-conserving discontinuous Galerkin (DG) method in [18] for one-dimensional linear hyperbolic equations. We prove the approximate solution superconverges to a particular projection of the exact solution. The order of this superconvergence is proved to be $k+2$ when piecewise $\mathbb {P}^k$ polynomials with $k \ge 1$ are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise $\mathbb {P}^k$ polynomials with arbitrary $k \ge 1$. Furthermore, we find that the derivative and function value approximations of the DG solution are superconvergent at a class of special points, with an order of $k+1$ and $k+2$, respectively. We also prove, under suitable choice of initial discretization, a ($2k+1$)-th order superconvergence rate of the DG solution for the numerical fluxes and the cell averages. Numerical experiments are given to demonstrate these theoretical results.