Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T14:38:25.821Z Has data issue: false hasContentIssue false
  • English
  • Français

A Reconstructed Discontinuous Galerkin Method for the Euler Equations on Arbitrary Grids

Published online by Cambridge University Press:  20 August 2015

Hong Luo ,
Luqing Luo  and
Robert Nourgaliev
Hong Luo*
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, 27695, USA
Luqing Luo*
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, 27695, USA
Robert Nourgaliev*
Affiliation:
Thermal Science and Safety Analysis Department, Idaho National Laboratory, Idaho Falls, ID, 83415, USA
*
Corresponding author.Email address:hong_luo@ncsu.edu
Email address:lluo2@ncsu.edu
Email address:Robert.Nourgaliev@inl.gov
Get access

Abstract

A reconstruction-based discontinuous Galerkin (RDG(P1P2)) method, a variant of P1P2 method, is presented for the solution of the compressible Euler equations on arbitrary grids. In this method, an in-cell reconstruction, designed to enhance the accuracy of the discontinuous Galerkin method, is used to obtain a quadratic polynomial solution (P2) from the underlying linear polynomial (P1) discontinuous Galerkin solution using a least-squares method. The stencils used in the reconstruction involve only the von Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG(P1P2) method is third-order accurate, and outperforms the third-order DG method (DG(P2)) in terms of both computing costs and storage requirements.

Keywords

65M6065M9976M2576M10Discontinuous Galerkin methodsleast-squares reconstructioncompressible Euler equations
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 5 , November 2012 , pp. 1495 - 1519
Copyright
Copyright © Global Science Press Limited 2012

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. and 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., and Shu, C.W., TVD Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for conservation laws IV: The Multidimensional Case, Mathematics of Computation, Vol. 55, pp. 545–581,1990.Google Scholar
[3]Cockburn, B. and Shu, C.W., The Runge-Kutta Discontinuous Galerkin Method for conservation laws V: Multidimensional System, Journal of Computational Physics, Vol. 141, pp. 199–224,1998.Google Scholar
[4]Cockburn, B., Karniadakis, G., and Shu, C.W., The Development of Discontinuous Galerkin Method, in Discontinuous Galerkin Methods, Theory, Computation, and Applications, edited by Cockburn, B., Karniadakis, G.E., and Shu, C.W., Lecture Notes in Computational Science and Engineering, Springer-Verlag, New York, 2000, Vol. 11 pp. 5–50, 2000.Google Scholar
[5]Bassi, F. and Rebay, S., High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations, Journal of Computational Physics, Vol. 138, pp. 251–285,1997.Google Scholar
[6]Atkins, H.L. and Shu, C.W., Quadrature Free Implementation of Discontinuous Galerkin Method for Hyperbolic Equations, AIAA Journal, Vol. 36, No. 5, 1998.CrossRefGoogle Scholar
[7]Bassi, F. and Rebay, S., GMRES discontinuous Galerkin solution of the Compressible Navier-Stokes Equations, Discontinuous Galerkin Methods, Theory, Computation, and Applications, edited by Cockburn, B., Karniadakis, G.E., and Shu, C.W., Lecture Notes in Computational Science and Engineering, Springer-Verlag, New York, 2000, Vol. 11, pp. 197–208,2000.Google Scholar
[8]Warburton, T.C. and Karniadakis, G. E., A Discontinuous Galerkin Method for the Viscous MHD Equations, Journal of Computational Physics, Vol. 152, pp. 608–641,1999.CrossRefGoogle Scholar
[9]Hesthaven, J.S. and Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Texts in Applied Mathematics, Vol. 56, 2008.CrossRefGoogle Scholar
[10]Rasetarinera, P. and Hussaini, M.Y., An Efficient Implicit Discontinuous Spectral Galerkin Method, Journal of Computational Physics, Vol. 172, pp. 718–738, 2001.Google Scholar
[11]Helenbrook, B.T., Mavriplis, D., and Atkins, H.L., Analysis of p-Multigrid for Continuous and Discontinuous Finite Element Discretizations, AIAA Paper 2003–3989, 2003.Google Scholar
[12]Fidkowski, K.J., Oliver, T.A., Lu, J., and Darmofal, D.L., p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations, Journal of Computational Physics, Vol. 207, No. 1, pp. 92–113,2005.Google Scholar
[13]Luo, H., Baum, J.D., and Lohner, R., A Discontinuous Galerkin Method Using Taylor Basis for Compressible Flows on Arbitrary Grids, Journal of Computational Physics, Vol. 227, No 20, pp. 8875–8893, October 2008.Google Scholar
[14]Luo, H., Baum, J.D., and Lohner, R., On the Computation of Steady-State Compressible Flows Using a Discontinuous Galerkin Method, International Journal for Numerical Methods in Engineering, Vol. 73, No. 5, pp. 597–623, 2008.Google Scholar
[15]Luo, H., Baum, J. D., and Lohner, R., A Hermite WENO-based Limiter for Discontinuous Galerkin Method on Unstructured Grids, Journal of Computational Physics, Vol. 225, No. 1, pp. 686–713,2007.Google Scholar
[16]Luo, H., Baum, J.D., and Lohner, R., A p-Multigrid Discontinuous Galerkin Method for the Euler Equations on Unstructured Grids, Journal of Computational Physics, Vol. 211, No. 2, pp. 767–783, 2006.Google Scholar
[17]Luo, H., Baum, J.D., and Lohner, R., Fast, p-Multigrid Discontinuous Galerkin Method for Compressible Flows at All Speeds, AIAA Journal, Vol. 46, No. 3, pp.635–652, 2008.CrossRefGoogle Scholar
[18] M. Dumbser, Balsara, D.S., Toro, E.F., and Munz, C.D., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, Journal of Computational Physics, Vol 227, pp. 8209–8253,2008.Google Scholar
[19]Dumbser, M. and Zanotti, O., Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations, Journal of Computational Physics, Vol. 228, pp. 6991–7006,2009.Google Scholar
[20]Dumbser, M., Arbitrary High Order PNPM Schemes on Unstructured Meshes for the Compressible Navier-Stokes Equations, Computers & Fluids, Vol. 39, pp. 60–76. 2010.Google Scholar
[21]Bassi, F. and Rebay, S., A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations, Journal of Computational Physics, Vol. 131, pp. 267–279,1997.Google Scholar
[22]Bassi, F. and Rebay, S., Discontinuous Galerkin Solution of the Reynolds-Averaged Navier-Stokes and k-w Turbulence Model Equations, Journal of Computational Physics, Vol. 34, pp. 507–540,2005.Google Scholar
[23]Cockburn, B. and Shu, C.W., The Local Discontinuous Galerkin Method for Time-dependent Convection-Diffusion System, SIAM Journal of Numerical Analysis, Vol. 35, pp. 2440–2463, 1998.Google Scholar
[24]Baumann, C.E. and Oden, J.T., A Discontinuous hp Finite Element Method for the Euler and Navier-Stokes Equations, International Journal for Numerical Methods in Fluids, Vol. 31, pp. 79–95,1999.Google Scholar
[25]Peraire, J. and Persson, P.O., The Compact Discontinuous Galerkin Method for Elliptic Problems. SIAM Journal on Scientific Computing, Vol. 30, pp. 1806–1824,2008.Google Scholar
[26]Arnold, D.N., Brezzi, F., Cockburn, B., and Marini, L.D., Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems, SIAM Journal on Numerical Analysis. Vol. 39, No. 5, pp. 1749–1779,2002.Google Scholar
[27]Gassner, G., Lorcher, F., and Munz, C.D., A Contribution to the Construction of Diffusion Fluxes for Finite Volume and Discontinuous Galerkin Schemes, Journal of Computational Physics, Vol. 224, No. 2, pp. 1049–1063,2007.CrossRefGoogle Scholar
[28]Liu, H. and Xu, K., A Runge-Kutta Discontinuous Galerkin Method for Viscous Flow Equations, Journal of Computational Physics, Vol. 224, No. 2, pp. 1223–1242,2007.CrossRefGoogle Scholar
[29]Luo, H., Luo, L., and Xu, K., A Discontinuous Galerkin Method Based on a BGK Scheme for the Navier-Stokes Equations on Arbitrary Grids, Advances in Applied Mathematics and Mechanics, Vol. 1, No. 3, pp. 301–318,2009.Google Scholar
[30]Leer, B. van and Nomura, S., Discontinuous Galerkin Method for Diffusion, AIAA Paper 2005–5108,2005.Google Scholar
[31]Leer, B. van and Lo, M., A Discontinuous Galerkin Method for Diffusion Based on Recovery, AIAA Paper 2007–4083, 2007.Google Scholar
[32]Raalte, M. and Leer, B. van, Bilinear Forms for the Recovery-Based Discontinuous Galerkin Method for Diffusion, Communication in Computational Physics, Vol. 5, No. 2-4, pp. 683693, 2009.Google Scholar
[33]Nourgaliev, R., Park, H., and Mousseau, V., Recovery Discontinuous Galerkin Jacobian-Free Newton-Krylov Method for Multiphysics Problems, Computational Fluid Dynamics Review 2010, 2010.Google Scholar
[34]Colella, P. and Woodward, P.R., The Piecewise Parabolic Method (PPM) for Gas-Dynamical Simulations, Journal of Computational Physics, Vol. 54, No. 1, pp. 115–173,1984.Google Scholar
[35]Luo, H., Luo, L., Nourgaliev, R., Mousseau, V., and Dinh, N., A Reconstructed Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations on Arbitrary Grids, Journal of Computational Physics, Vol. 229, pp. 6961–6978,2010.Google Scholar
[36]Aftosmis, M., Gaitonde, D., and Tavares, T.S., On the Accuracy, Stability, and Monotonicity of Various Reconstruction Algorithms for Unstructured Meshes, AIAA Paper 94–0415,1994.Google Scholar
[37]Haselbacher, A., McGuirk, J.J., and Page, G.J., Finite-Volume Discretization Aspects for Viscous Flows for Mixed Unstructured Meshes, AIAA Journal, 37(2), pp. 177–184,1999.CrossRefGoogle Scholar
[38]Luo, H., Luo, L., Ali, A., Nourgaliev, R., and Cai, C., A Parallel, Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Arbitrary Grids, Communication in Computational Physics, Vol. 9, pp. 363–389,2010.Google Scholar
[39]Luo, H., Sharov, D., Baum, J.D., and Lohner, R., On the Computation of Compressible Turbulent Flows on Unstructured Grids, International Journal of Computational Fluid Dynamics, Vol. 14, pp. 253–270, 2001.Google Scholar
[40]Luo, H., Baum, J.D., and Lohner, R., A Fast, Matrix-free Implicit Method for Compressible Flows on Unstructured Grids, Journal of Computational Physics, Vol. 146, No. 2, pp. 664690,1998.Google Scholar
[41]Luo, H., Baum, J.D., and Lohner, R., High-Reynolds Number Viscous Flow Computations Using an Unstructured-Grid Method, Journal of Aircraft, Vol. 42, No. 2, pp. 483–492, 2005.Google Scholar
[42]Luo, H., Baum, J.D., and Lohner, R., Extension of HLLC Scheme for Flows at All Speeds, AIAA Journal, Vol. 43, No. 6, pp. 1160–1166,2005.Google Scholar
[43]Zhang, L.P., Liu, W., He, L.X., Deng, X.G., and Zhang, H.X., A Class of Hybrid DG/FV Methods for Conservation Laws II: Two-dimensional Cases, Journal of Computational Physics, Vol. 231, pp. 1104–1120,2011.Google Scholar
[44]Sharov, D., Luo, H., Baum, J.D., and Lohner, R., Unstructured Navier-Stokes Grid Generation at Corners and Ridges, International Journal for Numerical Methods in Fluids, Vol. 43, pp. 717–728,2003.CrossRefGoogle Scholar
[45]Luo, H., Spiegel, S., and Lohner, R., A Hybrid Unstructured Cartesian and Triangular/Tetrahedral Grid Generation Method for Complex Geometries, AIAA journal. Vol. 48, No., 11 pp. 2639–2647,2010.Google Scholar
[46]Abgrall, R., On Essentially Non-Oscillatory Schemes on Unstructured Meshes: Analysis and Implementation, Journal of Computational Physics, Vol. 114, pp. 45–58,1994.CrossRefGoogle Scholar
[47]Friedrich, O., Weighted Essentially Non-Oscillatory Schemes for the Interpolation of Mean Values on Unstructured Grid, Journal of Computational Physics, Vol. 144, pp. 194–212,1998.Google Scholar
[48]Dumbser, M. and Kaser, M., Arbitrary High Order Non-Oscillatory Finite Volume Schemes on Unstructured Meshes for Linear Hyperbolic Systems, Journal of Computational Physics, Vol. 221, pp. 693–723,2007.Google Scholar
[49]Dumbser, M., Kaser, M., Titarev, V.A., and Toro, E.F., Quadrature-free Non-Oscillatory Finite Volume Schemes on Unstructured Meshes for Nonlinear Hyperbolic Systems, Journal of Computational Physics, Vol. 226, pp. 204–243, 2007.CrossRefGoogle Scholar
[50]Hu, C. and Shu, C.W., Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes, Journal of Computational Physics, Vol. 150, pp. 97–127,1999.Google Scholar
[51]Zhang, Y.T. and Shu, C.W., Third Order WENO Scheme on Three Dimensional Tetrahedral Meshes, Communication in Computational Physics, Vol. 5, pp. 836–848, 2009.Google Scholar
[52]Tsoutsanis, P., Titarev, V.A., and Drikakis, D., WENO Schemes on Arbitrary Mixed-Element Unstructured Meshes in Three Space Dimensions, Journal of Computational Physics, Vol. 230, pp. 1585–1601,2011.CrossRefGoogle Scholar