Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T03:10:23.229Z Has data issue: false hasContentIssue false
  • English
  • Français

A Parallel, High-Order Direct Discontinuous Galerkin Method for the Navier-Stokes Equations on 3D Hybrid Grids

Part of: Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  27 March 2017

Jian Cheng ,
Xiaodong Liu ,
Tiegang Liu  and
Hong Luo
Jian Cheng*
Affiliation:
School of Mathematics and Systems Science, Beihang University, Beijing 100191, P.R. China
Xiaodong Liu*
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, 27695, USA
Tiegang Liu*
Affiliation:
School of Mathematics and Systems Science, Beihang University, Beijing 100191, P.R. China
Hong Luo*
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, 27695, USA
*
*Corresponding author. Email addresses:[email protected] (J. Cheng), [email protected] (X. Liu), [email protected] (T. Liu), [email protected] (H. Luo)
*Corresponding author. Email addresses:[email protected] (J. Cheng), [email protected] (X. Liu), [email protected] (T. Liu), [email protected] (H. Luo)
*Corresponding author. Email addresses:[email protected] (J. Cheng), [email protected] (X. Liu), [email protected] (T. Liu), [email protected] (H. Luo)
*Corresponding author. Email addresses:[email protected] (J. Cheng), [email protected] (X. Liu), [email protected] (T. Liu), [email protected] (H. Luo)
Get access

Abstract

A parallel, high-order direct Discontinuous Galerkin (DDG) method has been developed for solving the three dimensional compressible Navier-Stokes equations on 3D hybrid grids. The most distinguishing and attractive feature of DDG method lies in its simplicity in formulation and efficiency in computational cost. The formulation of the DDG discretization for 3D Navier-Stokes equations is detailed studied and the definition of characteristic length is also carefully examined and evaluated based on 3D hybrid grids. Accuracy studies are performed to numerically verify the order of accuracy using flow problems with analytical solutions. The capability in handling curved boundary geometry is also demonstrated. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results obtained indicate that the DDG method can achieve the designed order of accuracy and is able to deliver comparable results as the widely used BR2 scheme, clearly demonstrating that the DDG method provides an attractive alternative for solving the 3D compressible Navier-Stokes equations.

Keywords

Direct discontinuous Galerkin methodcompressible Navier-Stokes equationshybrid grids

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M99: None of the above, but in this section 35L65: Conservation laws
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 5 , May 2017 , pp. 1231 - 1257
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Reed, W.H., Hill, T.R., Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Laboratory Report, LA-UR-73-479, 1973.Google Scholar
[2] Cockburn, B., Hou, S., Shu, C.-W., TVD Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Mathematics of Computation, 55(1990), pp. 545–581.Google Scholar
[3] Cockburn, B., Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional system, J. Comput. Phys., 141(1998), pp. 199–224.Google Scholar
[4] Luo, H., Baum, J.D., Löhner, R., A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids, J. Comput. Phys., 227(2008), pp. 8875–8893.CrossRefGoogle Scholar
[5] Luo, H., Luo, L.Q., Nourgaliev, R., Mousseau, V.A., Dinh, N., A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids, J. Comput. Phys., 229(2010), pp. 6961–6978.Google Scholar
[6] van Leer, B., Nomura, S., Discontinuous Galerkin method for Diffusion, AIAA-2005-5108.Google Scholar
[7] van Leer, B., Lo, M., A discontinuous Galerkinmethod for diffusion based on recovery, AIAA-2007-4083.Google Scholar
[8] Liu, H.L., Yan, J., The direct discontinuous Galerkin (DDG) methods for diffusion problems SIAM J. Numer. Anal., 47(2009), pp. 675–698.CrossRefGoogle Scholar
[9] Liu, H.L., Yan, J., The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections, Commun. Comput. Phys., 8(2010), pp. 541–564.Google Scholar
[10] Kannan, R., Wang, Z.J., The direct discontinuous Galerkin (DDG) viscous flux scheme for the high order spectral volume method, Computers & Fluids, 39(2010), pp. 2007–2021.CrossRefGoogle Scholar
[11] Bassi, F., Rebay, S., Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k–w turbulence model equations, Computers & Fluids, 34(2005), pp. 507–540.CrossRefGoogle Scholar
[12] Cockburn, B., Shu, C.-W., The local discontinuous Galerkin method for time dependent convection-diffusion systems, SIAM J. Numer. Anal., 35(1998), pp. 2440–2463.CrossRefGoogle Scholar
[13] Gao, H.Y., Wang, Z.J., Huynh, H.T., Differential formulation of discontinuous Galerkin and related methods for the Navier-Stokes equations, Commun. Comput. Phys., 13(2013), pp. 1013–1044.Google Scholar
[14] Peraire, J., Persson, P.O., The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J. Sci. Comput., 30(2008), pp. 1806–1824.CrossRefGoogle Scholar
[15] Bassi, F., Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of compressible Navier-Stokes equations, J. Comput. Phys., 131(1997), pp. 267–279.CrossRefGoogle Scholar
[16] Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D., Unified analysis of discontinuous Galerkin methods fro elliptic problems, SIAM J. Numer. Anal., 19(2002), pp. 742–760.CrossRefGoogle Scholar
[17] Hartmann, R., Houston, P., An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations, J. Comput. Phys., 227(2008), pp. 9670–9685.Google Scholar
[18] Douglas, J., Dupont, T., Interior penalty procedures for eplliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys., Springer, Berlin, 1976, pp. 207–216.Google Scholar
[19] Arnold, D.N., An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19(1982), pp. 742–760.CrossRefGoogle Scholar
[20] Wheeler, M.F., An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal., 15(1978), pp. 152–161.Google Scholar
[21] Gassner, G., Löcher, F., Munz, C.-D., A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes, J. Comput. Phys., 224(2207), pp. 1049–1063.Google Scholar
[22] Luo, H., Baum, J.D., Löhner, R., A fast, matrix-free implicit method for compressible flows on unstructured grids, J. Comput. Phys., 146(1998), pp. 664–690.Google Scholar
[23] Sun, Y., Wang, Z.J., Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids: extension to viscous flow, J. Comput. Phys., 215(2006), pp. 41–58.CrossRefGoogle Scholar
[24] Liu, Y., Vinokur, M., Wang, Z.J., Spectral (finite) volume method for conservation laws on unstructured grids V: extension to three-dimensional systems, J. Comput. Phys., 212(2006), pp. 454–472.CrossRefGoogle Scholar
[25] Luo, H., Xia, Y.D., Li, S.J., Nourgaliev, R., Cai, C.P., A Hermite WENO reconstruction-based discontinuous Galerkin method for the Euler equations on tetrahedral grids, J. Comput. Phys., 231(2012), pp. 5489–5503.Google Scholar
[26] Luo, H., Xia, Y.D., Spiegel, S., Nourgaliev, R., Jiang, Z.G., A reconstructed discontinuous Galerkin method based on a Hierarchical WENO reconstruction for compressible flows on tetrahedral grids, J. Comput. Phys., 236(2013), pp. 477–492.Google Scholar
[27] Cockburn, B., Gopalakrishnan, J., Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47(2009), pp. 1319–1365.Google Scholar
[28] Nguyen, N.C., Peraire, J., Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys., 228(2009), pp. 3232–3254.Google Scholar
[29] Kroll, N., ADGIMA, A European project on the development of adaptive higher-order variational methods for aerospace applications, AIAA-2009-176.Google Scholar
[30] The third international workshop on high-order CFD methods, https://www.grc.nasa.gov/hiocfd/.Google Scholar
[31] Leicht, T., Hartmann, R., Error estimation and anisotropic mesh refinement for 3D laminar aerodynamic flow simulations, J. Comput. Phys., 229(2010), pp. 7344–7360.Google Scholar
[32] Cheng, J., Yang, X.Q., Liu, T.G., Luo, H., A direct discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids, AIAA-2016-1344.CrossRefGoogle Scholar
[33] Karypis, G., Kumar, V., Metis-unstructured graph partitioning and sparse matrix ordering system, version 2.0, 1995.Google Scholar
[34] Yang, X.Q., Cheng, J., Wang, C.J., Luo, H., Si, J.T., A fast, implicit discontinuous Galerkin method based on analytical Jacobians for the compressible Navier-Stokes equations, AIAA-2016-1326.Google Scholar
[35] Castonguay, P., High-order energy stable flux reconstruction schemes for fluid flow simulations on unstructured grids, Ph.D. thesis, Stanford University, 2012.Google Scholar
[36] Liu, H.L., Optimal error estimates of the direct discontinuous Galerkin method for convection-diffusion equations, Math. Comp., 84(2015), pp. 2263–2295.Google Scholar
[37] Cao, W.-X., Liu, H.L., Zhang, Z.-M., Superconvergence of the direct discontinuous Galerkin method for convection-diffusion equations, Numer. Methods Partial Differential Eq., doi:10.1002/num.22087, 2016.Google Scholar