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

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

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

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 $O\left(h^{k+2}\right)$, when piecewise polynomials of degree $k$ are used. We then prove that the highest superconvergence rate of the DG solution itself is $O\left(h^{k+3/2}\right)$ 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.

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

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