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

Cao, Waixiang; Shu, Chi-Wang; Zhang, Zhimin

doi:10.1051/m2an/2017026

Cao, Waixiang ¹ ; Shu, Chi-Wang ² ; Zhang, Zhimin ^3,⁴

¹ School of Data and Computer Science, Sun Yat-sen University, Guangzhou 510006, China.
² Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
³ Beijing Computational Science Research Center, Beijing 100193, China.
⁴ Department of Mathematics, Wayne State University, Detroit, MI 48202, USA.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2213-2235.

Résumé

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.

Reçu le : 2016-08-25
Accepté le : 2017-04-26

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/m2an/2017026

Classification : 65M15, 65M60, 65N30
Mots-clés : Discontinuous Galerkin methods, superconvergence, degenerate variable coefficients, Radau points, upwind fluxes

Affiliations des auteurs :

Cao, Waixiang ¹ ; Shu, Chi-Wang ² ; Zhang, Zhimin ^{3, 4}

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     title = {Superconvergence of discontinuous {Galerkin} methods for {1-D} linear hyperbolic equations with degenerate variable coefficients},
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     pages = {2213--2235},
     publisher = {EDP-Sciences},
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Cao, Waixiang; Shu, Chi-Wang; Zhang, Zhimin. Superconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2213-2235. doi : 10.1051/m2an/2017026. http://www.numdam.org/articles/10.1051/m2an/2017026/

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