An all Mach number relaxation upwind scheme
The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 1-31.

The present paper concerns the derivation of finite volume methods to approximate the weak solutions of the Euler equations within all Mach number regimes. To address such an issue, we develop a Suliciu relaxation type scheme. By adopting a relevant scaling according to the Mach number, the obtained numerical scheme is proved to be accurate in the sense that the numerical viscosity does not increase as soon as the Mach number tends to zero. Moreover, the obtained scheme is proved to be asymptotic preserving since the correct incompressible asymptotic regime is recovered in the limit of the Mach number to zero. In addition, the robustness of the method is established since both density and internal energy remain positive during the simulations. Several numerical experiments in 1D and 2D are performed to illustrate the relevance of the proposed low Mach number numerical scheme.

Published online:
DOI: 10.5802/smai-jcm.60
Classification: 65M60, 65M12
Keywords: Hyperbolic system; Euler flow; Low Mach number flows; asymptotic preserving schemes; relaxation schemes; upwind schemes
Christophe Berthon 1; Christian Klingenberg 2; Markus Zenk 2

1 Université de Nantes, CNRS UMR 6629, Laboratoire de Mathématiques Jean Leray, 2 rue de la Houssinière, BP 92208, 44322 Nantes, France.
2 Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Strasse 30, 97074 Würzburg, Germany.
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{SMAI-JCM_2020__6__1_0,
     author = {Christophe Berthon and Christian Klingenberg and Markus Zenk},
     title = {An all {Mach} number relaxation upwind scheme},
     journal = {The SMAI Journal of computational mathematics},
     pages = {1--31},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {6},
     year = {2020},
     doi = {10.5802/smai-jcm.60},
     mrnumber = {4100530},
     zbl = {07207992},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.60/}
}
TY  - JOUR
AU  - Christophe Berthon
AU  - Christian Klingenberg
AU  - Markus Zenk
TI  - An all Mach number relaxation upwind scheme
JO  - The SMAI Journal of computational mathematics
PY  - 2020
SP  - 1
EP  - 31
VL  - 6
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.60/
DO  - 10.5802/smai-jcm.60
LA  - en
ID  - SMAI-JCM_2020__6__1_0
ER  - 
%0 Journal Article
%A Christophe Berthon
%A Christian Klingenberg
%A Markus Zenk
%T An all Mach number relaxation upwind scheme
%J The SMAI Journal of computational mathematics
%D 2020
%P 1-31
%V 6
%I Société de Mathématiques Appliquées et Industrielles
%U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.60/
%R 10.5802/smai-jcm.60
%G en
%F SMAI-JCM_2020__6__1_0
Christophe Berthon; Christian Klingenberg; Markus Zenk. An all Mach number relaxation upwind scheme. The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 1-31. doi : 10.5802/smai-jcm.60. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.60/

[1] W. Barsukow; P. Edelmann; C. Klingenberg; F. Miczek; F. Röpke A numerical scheme for the compressible low-Mach number regime of idela fluid dynamics, J. Sci. Comput., Volume 72 (2017) no. 2, pp. 623-646 | DOI | Zbl

[2] M. Baudin; C. Berthon; F. Coquel; R. Masson; Q. H. Tran A relaxation method for two-phase flow models with hydrodynamic closure law, Numer. Math., Volume 99 (2005) no. 3, pp. 411-440 | DOI | MR | Zbl

[3] G. Bispen; K. R. Arun; M. Lukáčová-Medvidóvá; S. Noelle IMEX large time step finite volume methods for low Froude number shallow water flows, Commun. Comput. Phys., Volume 16 (2014) no. 2, pp. 307-347 | DOI | MR | Zbl

[4] S. Boscarino; L. Pareschi; G. Russo Implicit-Explicit Runge–Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit, SIAM J. Sci. Comput., Volume 35 (2013) | DOI | MR | Zbl

[5] F. Bouchut Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Birkhäuser, 2004 | Zbl

[6] C. Chalons; F. Coquel; E. Godlewski; P.-A. Raviart; N. Seguin Godunov-type schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction, Math. Models Methods Appl. Sci., Volume 20 (2010) no. 11, pp. 2109-2166 | DOI | MR | Zbl

[7] G. Q. Chen; C. D. Levermore; T.-P. Liu Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. Pure Appl. Math., Volume 47 (1994), pp. 787-830 | DOI | MR | Zbl

[8] F. Coquel; B. Perthame Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics, SIAM J. Numer. Anal., Volume 35 (1998) no. 6, pp. 2223-2249 | DOI | MR | Zbl

[9] A. De Coninck; B. De Baets; D. Kourounis; F. Verbosio; O. Schenk; S. n Maenhout; J. Fostier Needles: Toward Large-Scale Genomic Prediction with Marker-by-Environment Interaction, Genetics, Volume 203 (2016) no. 1, pp. 543-555 | DOI

[10] P. Degond; M. Tang All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations, Commun. Comput. Phys., Volume 10 (2011) no. 1, pp. 1-31 | DOI | MR | Zbl

[11] S. Dellacherie Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number, J. Comput. Phys., Volume 229 (2010) no. 4, pp. 978-1016 | DOI | MR | Zbl

[12] G. Dimarco; L. Pareschi Asymptotic Preserving Implicit-Explicit Runge–Kutta Methods for Nonlinear Kinetic Equations, SIAM J. Numer. Anal., Volume 51 (2013) | DOI | MR | Zbl

[13] E. Feireisl; C. Klingenberg; S. Markfelder On the low Mach number limit for the compressible Euler system (preprint, https://arxiv.org/abs/1804.09509) | DOI | Zbl

[14] E. Godlewski; P.-A. Raviart Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, 118, Springer, 1996 | MR | Zbl

[15] H. Guillard; A. Murrone On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes, Computers & Fluids, Volume 33 (2004), pp. 655-675 | DOI | Zbl

[16] H. Guillard; C. Viozat On the behaviour of upwind schemes in the low Mach number limit, Computers & Fluids, Volume 28 (1999), pp. 63-86 | DOI | MR | Zbl

[17] J. Haack; S. Jin; J. Liu An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equations, Commun. Comput. Phys., Volume 12 (2012) no. 4, pp. 955-980 | DOI | MR | Zbl

[18] A. Harten; P. D. Lax; B. Van Leer On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., Volume 25 (1983), pp. 35-61 | DOI | MR | Zbl

[19] S. Jin Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations, SIAM J. Sci. Comput., Volume 21 (1999) no. 2, pp. 441-454 | DOI | MR | Zbl

[20] C. A. Kennedy; M. H. Carpenter Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equation (2001) (Technical report)

[21] S. Klainerman; A. Majda Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Pure Appl. Math., Volume 34 (1981) no. 4, pp. 481-524 | DOI | MR | Zbl

[22] R. Klein Semi-implicit extension of a godunov-type scheme based on low mach number asymptotics I: One-dimensional flow, J. Comput. Phys., Volume 121 (1995) no. 2, pp. 213-237 | DOI | MR | Zbl

[23] R. Klein; N. Botta; T. Schneider; C. D. Munz; S. Roller; A. Meister; L. Hoffmann; T. Sonar Asymptotic adaptive methods for multi-scale problems in fluid mechanics, J. Eng. Math., Volume 39 (2001) no. 1, pp. 261-343 | DOI | MR | Zbl

[24] D. Kourounis; A. Fuchs; O. Schenk Towards the Next Generation of Multiperiod Optimal Power Flow Solvers, IEEE Trans. Power Syst., Volume 33 (2018) no. 4, pp. 4005-4014 | DOI

[25] F. Miczek Simulation of low Mach number astrophysical flows, Ph. D. Thesis, München, Technische Universität München, Diss. (2013)

[26] F. Miczek; F. Röpke; P. Edelmann A new numerical solver for flows at various Mach numbers, Astron. Astrophys., Volume 576 (2015) no. A50, p. 16

[27] S. Noelle; G. Bispen; K. R. Arun; M. Lukáčová-Medvidóvá; C. D. Munz An Asymptotic Preserving all Mach Number Scheme for the Euler Equations of Gas Dynamics, SIAM J. Sci. Comput., Volume 36 (2014) no. 6, pp. 989-1024 | DOI | Zbl

[28] B. Perthame; C. W. Shu On positivity preserving finite volume schemes for Euler equations, Numer. Math., Volume 73 (1996) no. 1, pp. 119-130 | DOI | MR | Zbl

[29] P. L. Roe Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., Volume 43 (1981) no. 2, pp. 357-372 | DOI | MR | Zbl

[30] G. A. Sod A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., Volume 27 (1978) no. 1, pp. 1-31 | DOI | MR | Zbl

[31] I. Suliciu On modelling phase transitions by means of rate-type constitutive equations, shock wave structure, Int. J. Eng. Sci., Volume 28 (1990), pp. 829-841 | DOI | MR | Zbl

[32] I. Suliciu Some stability-instability problems in phase transitions modelled by piecewise linear elastic or viscoelastic constitutive equations, Int. J. Eng. Sci., Volume 30 (1992), pp. 483-494 | DOI | MR | Zbl

[33] E. Turkel Preconditioned Methods for Solving the Incompressible and Low Speed Compressible Equations, J. Comput. Phys., Volume 72 (1987) | DOI | Zbl

[34] F. Verbosio; A. De Coninck; D. Kourounis; O. Schenk Enhancing the scalability of selected inversion factorization algorithms in genomic prediction, J. Comput. Sci., Volume 22 (2017) no. Supplement C, pp. 99-108 | DOI

[35] J. M. Weiss; W. A. Smith Preconditioning applied to variable and constant density flows, AIAA J., Volume 33 (1995) no. 11, pp. 2050-2057 | DOI | Zbl

[36] G. B. Whitham Linear and nonlinear waves, Pure and Applied Mathematics, John Wiley & Sons, 1974, xvi+636 pages | MR | Zbl

Cited by Sources: