Superconvergence of Discontinuous Galerkin Methods for Linear Hyperbolic Equations

Abstract. In this paper, we consider the discontinuous Galerkin (DG) methods to solve linear hyperbolic equations with singular initial data. With the help of weight functions, the superconvergence properties outside the pollution region will be investigated. We show that, by using piecewise polynomials of degree k and suitable initial discretizations, the DG solution is (2k+1)-th order accurate at the downwind points and (k + 2)-th order accurate at all the other downwindbiased Radau points. Moreover, the derivative of error between the DG and exact solutions converges at a rate of k + 1 at all the interior upwind-biased Radau points. Besides the above, the DG solution is also (k + 2)-th order accurate towards a particular projection of the exact solution and the numerical cell averages are (2k + 1)-th order accurate. Numerical experiments are presented to confirm the theoretical results.