Superconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients

¹ School of Data and Computer Science, Sun Yat-sen University, Guangzhou 510006, China.
caowx5@mail.sysu.edu.cn. Research supported in part by NSFC grant 11501026, and the China Postdoctoral Science Foundation grant 2016T90027.
² Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
shu@dam.brown.edu. Research supported in part by DOE grant DE-FG02-08ER25863 and NSF grant DMS-1418750.
³ Beijing Computational Science Research Center, Beijing 100193, China.
zmzhang@csrc.ac.cn
⁴ Department of Mathematics, Wayne State University, Detroit, MI 48202, USA.
zzhang@math.wayne.edu. Research supported in part by NSFC grants 11471031 and 91430216, NASF grant U1530401, and NSF grant DMS-1419040.

Received: 25 August 2016
Revised: 10 February 2017
Accepted: 26 April 2017

Abstract

In this paper, we study the superconvergence behavior of discontinuous Galerkin methods using upwind numerical fluxes for one-dimensional linear hyperbolic equations with degenerate variable coefficients. The study establishes superconvergence results for the flux function approximation as well as for the DG solution itself. To be more precise, we first prove that the DG flux function is superconvergent towards a particular flux function of the exact solution, with an order of O(h^k+2), when piecewise polynomials of degree k are used. We then prove that the highest superconvergence rate of the DG solution itself is O(h^k+3/2) as the variable coefficient degenerates or achieves the value zero in the domain. As byproducts, we obtain superconvergence properties for the DG solution and the DG flux function at special points and for cell averages. All theoretical findings are confirmed by numerical experiments.