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

摘要

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 Pk polynomials with K≥1 are used.The proof is valid for arbitrary non-uniform regular meshes and for piecewise polynomials with arbitrary K≥1.Furthermore,we find that the derivative and function value approxi?mations of the DG solution are superconvergent at a class of special points,with an order of k+1 and R+2,respectively.We also prove,under suitable choice of initial discretization,a(2k+l)-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.