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.

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