Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T03:35:00.959Z Has data issue: false hasContentIssue false
  • English
  • Français

An All-Regime Lagrange-Projection Like Scheme for the Gas Dynamics Equations on Unstructured Meshes

Part of: Basic methods in fluid mechanics Incompressible inviscid fluids Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems Compressible fluids and gas dynamics, general

Published online by Cambridge University Press:  22 June 2016

Christophe Chalons ,
Mathieu Girardin  and
Samuel Kokh
Christophe Chalons*
Affiliation:
LMV - UMR 8100, Univ. Versailles Saint-Quentin-en-Yvelines, UFR des Sciences, Bâtiment Fermat, 45 avenue des Etats-Unis, 78035 Versailles cedex, France
Mathieu Girardin*
Affiliation:
DEN/DANS/DM2S/STMF/LMEC CEA Saclay, bât. 454 PC 47, 91191 Gif sur Yvette Cedex, France LRC MANON, Laboratoire de Recherche Conventionné CEA/DEN/DANS/DM2S and UPMC-CNRS/LJLL
Samuel Kokh*
Affiliation:
Maison de la Simulation USR 3441, Digiteo Labs, bât. 565, PC 190, CEA Saclay, 91191 Gif-sur-Yvette, France DEN/DANS/DM2S/STMF, CEA Saclay, 91191 Gif-sur-Yvette, France
*
*Corresponding author. Email addresses:christophe.chalons@uvsq.fr (C. Chalons), mathieu.girardin@cea.fr (M. Girardin), samuel.kokh@cea.fr (S. Kokh)
*Corresponding author. Email addresses:christophe.chalons@uvsq.fr (C. Chalons), mathieu.girardin@cea.fr (M. Girardin), samuel.kokh@cea.fr (S. Kokh)
*Corresponding author. Email addresses:christophe.chalons@uvsq.fr (C. Chalons), mathieu.girardin@cea.fr (M. Girardin), samuel.kokh@cea.fr (S. Kokh)
Get access

Abstract

We propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics equations. By all regime, we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolved discretization with respect to the Mach number M, i.e. such that the ratio between the Mach number M and the mesh size or the time step is small with respect to 1. The key idea is to decouple acoustic and transport phenomenon and then alter the numerical flux in the acoustic approximation to obtain a uniform truncation error in term of M. This modified scheme is conservative and endowed with good stability properties with respect to the positivity of the density and the internal energy. A discrete entropy inequality under a condition on the modification is obtained thanks to a reinterpretation of the modified scheme in the Harten Lax and van Leer formalism. A natural extension to multi-dimensional problems discretized over unstructured mesh is proposed. Then a simple and efficient semi implicit scheme is also proposed. The resulting scheme is stable under a CFL condition driven by the (slow) material waves and not by the (fast) acoustic waves and so verifies the all regime property. Numerical evidences are proposed and show the ability of the scheme to deal with tests where the flow regime may vary from low to high Mach values.

Keywords

Gas dynamics equationslow-Mach regimefinite volume schemesall-regime schemesLagrange-Projection like schemeslarge time-steps

MSC classification

Secondary: 35L65: Conservation laws 35L40: First-order hyperbolic systems 65M08: Finite volume methods 76N15: Gas dynamics, general 76M12: Finite volume methods 76B99: None of the above, but in this section
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 1 , July 2016 , pp. 188 - 233
Copyright
Copyright © Global-Science Press 2016 

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] Bouchut, F.. Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources. Frontiers in Mathematics Series. Birkhäuser, 2004.CrossRefGoogle Scholar
[2] Chalons, C. and Coquel, F.. Navier-stokes equations with several independant pressure laws and explicit predictor-corrector schemes. Numerisch Math., 101(3):pp. 451–478, 2005.Google Scholar
[3] Chalons, C. and Coulombel, J.-F.. Relaxation approximation of the Euler equations. J. Math. Anal. Appl., 348(2):pp. 872–893, 2008.CrossRefGoogle Scholar
[4] Chalons, C., Girardin, M., and Kokh, S.. Large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms. SIAM J. Sci. Comput., 35(6):pp. a2874–a2902, 2013.Google Scholar
[5] Colella, Ph. and Pao, K.. A projection method for low speed flows. J. Comp. Phys., 149(2):pp. 245–269, 1999.Google Scholar
[6] Coquel, F., Nguyen, Q. L., Postel, M., and Tran, Q. H.. Entropy-satisfying relaxation method with large time-steps for Euler IBVPs. Math. Comput., 79(271):pp. 1493–1533, 2010.Google Scholar
[7] Cordier, F., Degond, P., and Kumbaro, A.. An Asymptotic-Preserving all-speed scheme for the Euler and NavierStokes equations. J. Comp. Phys., 231(17):pp. 5685–5704, 2012.Google Scholar
[8] Degond, P. and Tang, M.. All speed method for the Euler equation in the low mach number limit. Commun. Comp. Phys., 10:pp. 1–31, 2011.Google Scholar
[9] Degond, P., Jin, S., and Liu, J.-G.. Mach-number uniform asymptotic-preserving gauge schemes for compressible flows. Bull. Inst. Math., Acad. Sin. (N.S.), 2(4):pp. 851–892, 2007.Google Scholar
[10] Dellacherie, S., Omnes, P. and Raviart, P.A.. Construction of modified Godunov type schemes accurate at any Mach number for the compressible Euler system. submitted, 2013.Google Scholar
[11] Dellacherie, S.. Analysis of Godunov type schemes applied to the compressible euler system at low Mach number. J. Comp. Phys., 229(4):pp. 978–1016, 2010.CrossRefGoogle Scholar
[12] Després, B.. Inégalité entropique pour un solveur conservatif du système de la dynamique des gaz en coordonnées de lagrange. C. R. Acad. Sci. Paris, Série I, 324:pp. 1301–1306, 1997.Google Scholar
[13] Després, B., Labourasse, E., Lagoutière, F., and Marmajou, I.. An antidissipative transport scheme on unstructuredmeshes formulticomponent flows. Int. J. Finite. Vol. Meth., 7:pp. 30–65, 2010.Google Scholar
[14] Després, B.. Lois de Conservations Eulériennes, Lagrangiennes et Méthodes Numériques, volume 68 of Mathématiques et Applications, SMAI. Springer, 2010.CrossRefGoogle Scholar
[15] Dauvergne, F., Ghidaglia, J.-M., Pascal, F., and Rovarch, J.-M.. Renormalization of the numerical diffusion for an upwind finite volume method. application to the simulation of Kelvin-Helmholtz instability. Finite Volumes for Complex Applications. V. Proceedings of the 5th International Symposium, Aussois, June 2008, R. Eymard and J.-M. Hérard editors, pp. 321–328, 2008.Google Scholar
[16] De Vuyst, F. and Gasc, T.. Suitable formulations of Lagrange Remap Finite Volume schemes for manycore / GPU architectures. Finite Volumes for Complex Applications. VII. Proceedings of the 7th International Symposium, Berlin, June 2014, J. Fuhrmann, M. Ohlberger and Ch. Rohde editors, 2014.Google Scholar
[17] Girardin, M.. Asymptotic preserving and all-regime Lagrange-Projection like numerical schemes: Application to two-phase flows in low mach regime. Thèse de l’Université Pierre et Marie Curie Paris 6, 2014, available at https://tel.archives-ouvertes.fr/tel-01127428.Google Scholar
[18] Godlewski, E. and Raviart, P.-A.. Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, 1996.Google Scholar
[19] Guillard, H. and Viozat, C.. On the behavior of upwind schemes in the lowMach limit. Comp. & Fluid, 28:pp. 63–86, 1999.Google Scholar
[20] Harten, A., Lax, P.D., and Van Leer, B.. On upstream differencing and godunov-type schemes for hyperbolic conservation laws. SIAM Review, 25:pp. 35–61, 1983.Google Scholar
[21] Toumi, I., Kumbaro, A., and Paillere, H.. Approximate Riemann solvers and flux vector splitting schemes for two-phase flow. In VKI LS 1999-03, Computational Fluid Dynamics, 1999.Google Scholar
[22] Liu, J.-G. Haack, J., Jin, S.. An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equations. Commun. Comp. Phys., 12:pp. 955–980, 2012.Google Scholar
[23] Jin, S.. Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comp. Phys., 122(1):pp. 51–67, 1995.Google Scholar
[24] Klein, R.. Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: One-dimensional flow. J. Comp. Phys., 121(2):pp. 213–237, 1995.CrossRefGoogle Scholar
[25] Lax, P. D. and Liu, X.-D.. Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput., 19(2):pp. 319–340, 1998.Google Scholar
[26] Liou, M.-S.. A sequel to AUSM, part II: AUSM+-up for all speeds. J. Comp. Phys., 214(1):pp. 137–170, 2006.Google Scholar
[27] Paillère, H., Viozat, C., Kumbaro, A., and Toumi, I.. Comparison of low mach number models for natural convection problems. Heat and Mass Transfer, 36(6):pp. 567–573, 2000.Google Scholar
[28] Schochet, S.. Fast singular limits of hyperbolic PDEs. J. Differ. Equations, 114(2):pp. 476–512, 1994.Google Scholar
[29] Sod, G. A.. Numerical Methods in Fluid Dynamics. Initial and Initial-Boundary Value Problems. Cambridge: Cambridge University Press, 1985.CrossRefGoogle Scholar
[30] Suliciu, I.. On the thermodynamics of rate-type fluids and phase transitions. i. rate-type fluids. Int. J. Eng. Sci., 36(9):pp. 921–947, 1998.CrossRefGoogle Scholar
[31] Sun, M.. An implicit cell-centered Lagrange-Remap scheme for all speed flows. Computers & Fluids, Vol. 96, pp. 397–405, 2014.CrossRefGoogle Scholar
[32] Thornber, B., Mosedale, A., Drikakis, D., Youngs, D., and Williams, R.J.R.. An improved reconstruction method for compressible flows with low Mach number features. J. Comp. Phys., 227(10):pp. 4873–4894, 2008.Google Scholar
[33] Turkel, E.. Preconditioned methods for solving the incompressible and low speed compressible equations. J. Comp. Phys., 72(2):pp. 277–298, 1987.Google Scholar
[34] Weyl, H.. Shock waves in arbitrary fluids. Commun. Pure Appl. Math., 2:pp. 103–122, 1949.Google Scholar