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

Published online by Cambridge University Press:  03 June 2015

Haiyang Gao ,
Z. J. Wang  and
H. T. Huynh
Haiyang Gao*
Affiliation:
Department of Aerospace Engineering and CFD Center, Iowa State University, 2271 Howe Hall Ames, IA 50011, USA
Z. J. Wang*
Affiliation:
Department of Aerospace Engineering and CFD Center, Iowa State University, 2271 Howe Hall Ames, IA 50011, USA
H. T. Huynh*
Affiliation:
Department of Aerospace Engineering and CFD Center, Iowa State University, 2271 Howe Hall Ames, IA 50011, USA
*
Corresponding author.Email:hgao@iastate.edu
Email:zjw@iastate.edu
Email:hung.t.huynh@nasa.gov
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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 dif-fusion 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.

Keywords

76N15Discontinuous Galerkinlifting collocation penaltyflux reconstructionNavier-Stokes equationscorrection procedure via reconstructionunstructured hybrid grids
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 4 , April 2013 , pp. 1013 - 1044
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Arnold, D.N., Brezzi, F., Cockburn, B. and Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 19(4), 742–760 (2002).CrossRefGoogle Scholar
[2]Bassi, F. and Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131(1), 267–79 (1997).Google Scholar
[3]Bassi, F. and Rebay, S., High-order accurate discontinuous finite element solution of the 2D Euler equations, J. Comput. Phys., 138, 251–285 (1997).Google Scholar
[4]Bassi, F., Rebay, S., GMRES discontinuous Galerkin solution of the compressible Navier-Stokes equations, in: Cockburn, B., Karniadakis, G.E. and Shu, C.-W. (Eds.), Discontinuous Galerkin Methods: Theory, Computation and Applications, Springer, Berlin, 2000, pp. 197–208.Google Scholar
[5]Cockburn, B. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comp., 52, 411–435 (1989).Google Scholar
[6]Cockburn, B., Lin, S.-Y. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems, J. Comput. Phys., 84,90–113 (1989).Google Scholar
[7]Cockburn, B. and Shu, C.-W., The Runge-Kutta discontinuous Garlerkin method for conservation laws V: Multidimensional systems, J. Comput. Phys., 141,199–224 (1998).CrossRefGoogle Scholar
[8]Cockburn, B. and Shu, C.-W., The local discontinuous Galerkin methods for time-dependent convection diffusion systems, SIAM J. Numer. Anal., 35, 2440–2463 (1998).Google Scholar
[9]Dolejøí, V., On the discontinuous Galerkin method for numerical solution of the Navier-Stokes equations, Int. J. Numer. Methods Fluids, 45, 1083–1106 (2004).Google Scholar
[10]Gao, H. and Wang, Z.J., A high-order lifting collocation penalty formulation for the Navier-Stokes equations on 2-D mixed grids, AIAA Paper 2009–3784.Google Scholar
[11]Haga, T., Gao, H. and Wang, Z.J., A high-order unifying discontinuous formulation for 3-D mixed grids, AIAA Paper 2010–540.Google Scholar
[12]Hartmann, R. and Houston, P., Symmetric interior penalty DG methods for the incompressible Navier-Stokes Equations I: Method formulation, Int. J. Numer. Anal. Model., 3(1), 1–20 (2006).Google Scholar
[13]Huynh, H.T., A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, AIAA Paper 2007–4079.Google Scholar
[14]Huynh, H.T., A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion, AIAA Paper 2009–403.Google Scholar
[15]Kopriva, D.A. and Kolias, J.H., A conservative staggered-grid Chebyshev multidomain method for compressible flows, J. Comput. Phys., 125, 244–261 (1996).Google Scholar
[16]Liang, C., Jameson, A. and Wang, Z.J., Spectral difference method for two-dimensional compressible flow on unstructured grids with mixed elements, J. Comput. Phys., 228, 2847–2858 (2009).Google Scholar
[17]Liu, Y., Vinokur, M. and Wang, Z.J., Discontinuous spectral difference method for conservation laws on unstructured grids, in Proceedings of the 3rd International Conference on Computational Fluid Dynamics, Toronto, Canada, July 12–16 2004.Google Scholar
[18]Liu, Y., Vinokur, M. and Wang, Z.J., Spectral difference method for unstructured grids I: Basic formulation, J. Comput. Phys., 216, 780–801 (2006).CrossRefGoogle Scholar
[19]May, G. and Jameson, A., A spectral difference method for the Euler and Navier-Stokes equations, AIAA paper 2006–304.Google Scholar
[20]Peraire, J. and Persson, P.-O., The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J. Sci. Comput., 30(4), 1806–1824 (2008).Google Scholar
[21]Sun, Y., Wang, Z.J. and Liu, Y., Efficient implicit non-linear LU-SGS approach for compressible flow computation using high-order spectral difference method, Commun. Comput. Phys., 5(2–4), 760–778 (2009).Google Scholar
[22]den Abeele, K. Van, Lacor, C. and Wang, Z.J., On the stability and accuracy of the spectral difference method, J. Sci. Comput., 37(2), 162–188 (2008).Google Scholar
[23] B. van Leer and Nomura, S., Discontinuous Galerkin for diffusion, AIAA Paper 2005–5108.Google Scholar
[24]Wang, Z.J., Spectral (finite) volume method for conservation laws on unstructured grids: Basic formulation, J. Comput. Phys., 178, 210–251 (2002).Google Scholar
[25]Wang, Z.J., High-order methods for the Euler and Navier-Stokes equations on unstructured grids, J. Progress in Aerospace Sciences, 43,1–47 (2007).Google Scholar
[26]Wang, Z.J. and Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids III: One-dimensional systems and partition optimization, J. Sci. Comput., 20(1), 137157 (2004).CrossRefGoogle Scholar
[27]Wang, Z.J., Zhang, L. and Liu, Y., Spectral (finite) volume method for conservation laws on un-structured grids IV: Extension to two-dimensional Euler equations, J. Comput. Phys., 194(2), 716–741 (2004).Google Scholar
[28]Wang, Z.J. and Gao, H., A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids, J. Comput. Phys., 228, 8161–8186 (2009).Google Scholar
[29]Williamson, C.H.K., Oblique and parallel modes of vortex shedding in the wake of a cylinder at low Reynolds number, J. Fluid Mech., 206,579–627 (1989).Google Scholar