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 Pk polynomials with arbitrary k ≥ 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.
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