Commun. Comput. Phys., 13 (2013), pp. 1013-1044.


Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations

Haiyang Gao 1*, Z. J. Wang 1, H. T. Huynh 2

1 Department of Aerospace Engineering and CFD Center, Iowa State University, 2271 Howe Hall Ames, IA 50011, USA.
2 NASA Glenn Research Center, MS 5-11, Cleveland, OH 44135, USA.

Received 2 June 2011; Accepted (in revised version) 9 March 2012
Available online 21 September 2012
doi:10.4208/cicp.020611.090312a

Abstract

A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure. In addition to being simple and economical, it unifies several existing methods including discontinuous Galerkin, staggered grid, spectral volume, and spectral difference. To discretize the diffusion terms, we use the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG), and I-continuous approaches. The first three of these approaches, originally derived using the integral formulation, were recast here in the CPR framework, whereas the I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a triangular mesh. Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out. Finally, results for the Navier-Stokes equations are shown to compare the various schemes as well as to demonstrate the capability of the CPR approach.

AMS subject classifications: 76N15

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Key words: Discontinuous Galerkin, lifting collocation penalty, flux reconstruction, Navier-Stokes equations, correction procedure via reconstruction, unstructured hybrid grids.

*Corresponding author.
Email: hgao@iastate.edu (H. Gao), zjw@iastate.edu (Z. J. Wang), hung.t.huynh@nasa.gov (H. T. Huynh)
 

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