Superconvergence of Discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations^∗

¹ Beijing Computational Science Research Center, Zhongguancun Software Park II, No. 10 West Dongbeiwang Road, Haidian District, Beijing 100094, P.R. China.
² School of Mathematics and Statistics, Huazhong University of Science and Technology, 1037 Luoyu Rd, Hongshan, Wuhan, Hubei 430074, P.R. China.
³ Department of Mathematical Sciences, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, USA.
yyang7@mtu.edu
⁴ Department of Mathematics, Wayne State University, 42 W. Warren Ave. Detroit, MI 48202, USA.

Received: 20 January 2016
Revised: 24 March 2016
Accepted: 21 April 2016

Abstract

In this paper, we study superconvergence properties of the discontinuous Galerkin method using upwind-biased numerical fluxes for one-dimensional linear hyperbolic equations. A (2k + 1)th order superconvergence rate of the DG approximation at the numerical fluxes and for the cell average is obtained under quasi-uniform meshes and some suitable initial discretization, when piecewise polynomials of degree k are used. Furthermore, surprisingly, we find that the derivative and function value approximation of the DG solution are superconvergent at a class of special points, with an order k + 1 and k + 2, respectively. These superconvergent points can be regarded as the generalized Radau points. All theoretical findings are confirmed by numerical experiments.