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All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations

Published online by Cambridge University Press:  20 August 2015

Pierre Degond  and
Min Tang
Pierre Degond*
Affiliation:
Institute of Mathematics of Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France
Min Tang*
Affiliation:
Institute of Mathematics of Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France
*
Email:[email protected]
Corresponding author.Email:[email protected]
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Abstract

An all speed scheme for the Isentropic Euler equations is presented in this paper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficulty in the low Mach regime. The key idea of our all speed scheme is the special semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one being treated explicitly and the other one implicitly. Moreover, the flux of the density equation is also treated implicitly and an elliptic type equation is derived to obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods, firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs to be solved implicitly which reduces much computational cost. We develop this semi-implicit time discretization in the framework of a first order Local Lax-Friedrichs (or Rusanov) scheme and numerical tests are displayed to demonstrate its performances.

Keywords

65M0665Z0576N9976L05Low Mach numberIsentropic Euler equationscompressible flowincompressible limitasymptotic preservingRusanov scheme
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 1 , July 2011 , pp. 1 - 31
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Bijl, H. and Wesseling, P., A unified method for computing incompressible and compressible flows in boundary-fitted coordinates, J. Comput. Phys., 141 (1998), 153–173.CrossRefGoogle Scholar
[2]Bonner, M. P., Compressible Subsonic Flow on a Staggered Grid, master thesis, The University of British Columbia, (2007).Google Scholar
[3]Constantin, P., On the Euler equations of incompressible fluids, B. m. Math. Soc., 44(4) (2007), 603–621.Google Scholar
[4]Crispel, P., egond, P.and Vignal, M.-H., An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit, J. Comput. Phys., 223 (2007), 208–234.CrossRefGoogle Scholar
[5]Degond, P., Deluzet, F., Sangam, A. and Vignal, M-H., An asymptotic preserving scheme for the Euler equations in a strong magnetic field, J. Comput. Phys., 228(10) (2009), 3540–3558.CrossRefGoogle Scholar
[6]Degond, P., Jin, S. and Liu, J.-G., Mach-number uniform asymptotic-preserving gauge schemes for compressible flows, B. I. Math., Academia Sinica, New Series, 2(4) (2007), 851–892.Google Scholar
[7]Degond, P., Liu, J.-G. and Vignal, M.-H., Analysis of an asymptotic preserving scheme for the Euler-Poison system in the quasineutral limit, SIAM J. Numer. Anal., 46 (2008), 1298–1322.Google Scholar
[8]Degond, P., Peyrard, P. F., Russo, G. and Villedieu, P., Polynomial upwind schemes for hyperbolic systems, Partial Differential Equations, 1 (1999), 479–483.Google Scholar
[9]Golse, F., Jin, S. and Levermore, C. D., The convergence of numerical transfer schemes in diffusive regimes I: the discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333–1369.CrossRefGoogle Scholar
[10]Haack, J. R. and Hauck, C. D., Oscillatory behavior of asymptotic-preserving splitting methods for a linear model of diffusive relaxation, Los Alamos Report LA-UR 08-0571, appeared in Kinetic and Related Models, 2008.Google Scholar
[11]Haack, J., Jin, S. and Liu, J. G., All speed asymptotic preserving schemes for compressible flows, in preparation.Google Scholar
[12]Harlow, F. H. and Amsden, A., A numerical fluid dynamics calculation method for all flow speeds, J. Comput. Phys., 8 (1971), 197–213.Google Scholar
[13]Harlow, F. H. and Welch, J. E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluid., 8(12) (1965), 2182–2189.Google Scholar
[14]Issa, R. I., Gosman, A. D. and Watkins, A. P., The computation of compressible and incompressible flow of fluid with a free surface, Phys. Fluids., 8 (1965), 2182–2189.Google Scholar
[15]Klainerman, S. and Majda, A., Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun.Pure. Appl.Math., 34 (1981), 481–524.CrossRefGoogle Scholar
[16]Klainerman, S. and Majda, A., Compressible and incompressible fluids, Commun. Pure. Appl. Math., 35 (1982), 629–653.CrossRefGoogle Scholar
[17]Klein, R., Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: one-dimensional flow, J. Comput. Phys., 121 (1995), 213–237.CrossRefGoogle Scholar
[18]Klein, R., Botta, N., Schneider, T., Munz, C. D., Roller, S., Meister, A., Hoffmann, L. and Sonar, T., Asymptotic adaptive methods for multi-scale problems in fluid mechanics, J. Eng. Math., 83 (2001), 261–343.Google Scholar
[19]Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), 214–282.Google Scholar
[20]Kurganov, A. and Tadmor, E., Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numer. Meth. Part. D. E., 18 (2002), 548–608.Google Scholar
[21]Leveque, R. J., Numerical Methods for Conservation Laws, Lectures in Mathematics ETH Zürich, 1992.Google Scholar
[22]Leveque, R. J., Finite Volume Method for Hyperbolic Problems, Cambridge Text in Applied Mathematics, Cambridge University Press, 2002.Google Scholar
[23]Munz, C. D., Dumbser, M. and Roller, S., Linearized acoustic perturbation equations for low Mach number flow with variable density and temperature, J. Comput. Phys., 224 (2007), 352–364.CrossRefGoogle Scholar
[24]Munz, C. D., Roller, S., Klein, R. and Geratz, K. J., The extension of incompressible folw solvers to the weakly compressible regime, Comp. Fluid., 32 (2002), 173–196.Google Scholar
[25]Park, J. H. and Munz, C. D., Multiple pressure variables methods for fluid flow at all Mach numbers, Int. J. Numer. Meth. Fluid., 49 (2005), 905–931.CrossRefGoogle Scholar
[26]Patankar, S. V., Numerical Heat Transfer and Fluid Flow, New York: McGraw-Hill, 1980.Google Scholar
[27]Rieper, F. and Bader, G., The influence of cell geometry on the accuracy of upwind schemes in the low Mach number regime, J. Comput. Phys., 228 (2009), 2918–2933.Google Scholar
[28]Rusanov, V. V., Calculation of interaction of non-steady shock waves with obstacles, J. Comput. Math. Phys., 1 (1961), 267–279.Google Scholar
[29]Heul, D. R.van der, Vuik, C. and Wesseling, P., A conservative pressure-correction method for flow at all speeds, Comput. Fluid., 32 (2003), 1113–1132.Google Scholar