Commun. Comput. Phys., 5 (2009), pp. 456-468.


A Discontinuous Galerkin Extension of the Vertex-Centered Edge-Based Finite Volume Method

Martin Berggren 1, Sven-Erik Ekstrom 2*, Jan Nordstrom 2

1 Department of Computing Science, Umea University, SE-901 87 Umea Sweden.
2 Division of Scientific Computing, Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden.

Received 1 October 2007; Accepted (in revised version) 15 February 2008
Available online 1 August 2008

Abstract

The finite volume (FV) method is the dominating discretization technique for computational fluid dynamics (CFD), particularly in the case of compressible fluids. The discontinuous Galerkin (DG) method has emerged as a promising high-accuracy alternative. The standard DG method reduces to a cell-centered FV method at lowest order. However, many of today's CFD codes use a vertex-centered FV method in which the data structures are edge based. We develop a new DG method that reduces to the vertex-centered FV method at lowest order, and examine here the new scheme for scalar hyperbolic problems. Numerically, the method shows optimal-order accuracy for a smooth linear problem. By applying a basic hp-adaption strategy, the method successfully handles shocks. We also discuss how to extend the FV edge-based data structure to support the new scheme. In this way, it will in principle be possible to extend an existing code employing the vertex-centered and edge-based FV discretization to encompass higher accuracy through the new DG method.

AMS subject classifications: 65N22, 65N30 , 65N50

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Key words: Discontinuous Galerkin methods, finite volume methods, dual mesh, vertex-centered, edge-based, CFD.

*Corresponding author.
Email: martin.berggren@cs.umu.se (M. Berggren), sven-erik.ekstrom@it.uu.se (S.-E. Ekstrom), jan.nordstrom@it.uu.se (J. Nordstrom)
 

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