No Access

Construction of modified Godunov-type schemes accurate at any Mach number for the compressible Euler system

Commissariat à l’Énergie Atomique et aux Énergies Alternatives, CEA, DEN, DM2S, STMF, F-91191 Gif-sur-Yvette, France

Département de Génie Mécanique, École Polytechnique de Montréal, C. P. 6079, Succ. Centre-ville, Montréal H3C 3A7 (Québec), Canada

Université Pierre et Marie Curie (Paris 6), LRC-Manon, Laboratoire J. L. Lions, 4 Place Jussieu, 75005 Paris, France

E-mail Address: stephane.dellacherie@cea.fr

Search for more papers by this author

J. Jung

Université Pierre et Marie Curie (Paris 6), LRC-Manon, Laboratoire J. L. Lions, 4 Place Jussieu, 75005 Paris, France

Université de Pau et des Pays de l’Adour, LMA-IPRA, UMR CNRS 5142, Avenue de l’Université, 64013 Pau, France

INRIA Bordeaux Sud Ouest, Cagire Team, 351 Cours de la Libération, 33405 Talence, France

Search for more papers by this author

P. Omnes

Commissariat à l’Énergie Atomique et aux Énergies Alternatives, CEA, DEN, DM2S, STMF, F-91191 Gif-sur-Yvette, France

Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 99 Avenue J.-B. Clément F-93430, Villetaneuse Cedex, France

Search for more papers by this author

, and

P.-A. Raviart

Université Pierre et Marie Curie (Paris 6), LRC-Manon, Laboratoire J. L. Lions, 4 Place Jussieu, 75005 Paris, France

Search for more papers by this author

https://doi.org/10.1142/S0218202516500603Cited by:41 (Source: Crossref)

Abstract

This paper is composed of three self-consistent sections that can be read independently of each other. In Sec. 1, we define and analyze the low Mach number problem through a linear analysis of a perturbed linear wave equation. Then, we show how to modify Godunov-type schemes applied to the linear wave equation to make this scheme accurate at any Mach number. This allows to define an all Mach correction and to propose a linear all Mach Godunov scheme for the linear wave equation. In Sec. 2, we apply the all Mach correction proposed in Sec. 1 to the case of the nonlinear barotropic Euler system when the Godunov-type scheme is a Roe scheme. A linear stability result is proposed and a formal asymptotic analysis justifies the construction in this nonlinear case by showing how this construction is related with the linear analysis of Sec. 1. At last, we apply in Sec. 3 the all Mach correction proposed in Sec. 1 in the case of the full Euler compressible system. Numerous numerical results proposed in Secs. 1–3 justify the theoretical results and show that the obtained all Mach Godunov-type schemes are both accurate and stable for all Mach numbers. We also underline that the proposed approach can be applied to other schemes and allows to justify other existing all Mach schemes.

Communicated by F. Brezzi

Keywords:

AMSC: 35L05, 35L45, 35L65, 65M08, 76N99

References

1. F. Bassi, C. De Bartolo, R. Hartmann and A. Nigro, A discontinuous Galerkin method for inviscid low Mach number flows, J. Comput. Phys. 228 (2009) 3996–4011. Crossref, Web of Science, Google Scholar
2. M. Bernard, S. Dellacherie, G. Faccanoni, B. Grec and Y. Penel, Study of a low Mach nuclear core model for two-phase flows with phase transition I: Stiffened gas law, ESAIM Math. Model. Numer. Anal. 48 (2014) 1639–1679. Crossref, Web of Science, Google Scholar
3. F. Beux, M. V. Salvetti and E. Sinibaldi, A preconditioned implicit Roe’s scheme for barotropic flows: Towards simulation of cavitation phenomena, Technical Report 4891, INRIA (2003). Google Scholar
4. T. Buffard, Analyse de quelques méthodes de volumes finis non structurés pour la résolution des équations d’Euler, Ph.D. thesis, Université Pierre et Marie Curie (Paris) 6 (1994). Google Scholar
5. T. Buffard, T. Gallouët and J.-M. Hérard, A sequel to a rough Godunov scheme: Application to real gases, Comput. Fluids 29 (2000) 813–847. Crossref, Web of Science, Google Scholar
6. C. Chalons, M. Girardin and S. Kokh, An all-regime Lagrange-projection like scheme for the gas dynamics equations on unstructured meshes, preprint (2014), arXiv:hal-01007622. Google Scholar
7. S. Clerc, Numerical simulation of the homogeneous equilibrium model for two-phase flow, J. Comput. Phys. 161 (2000) 354–375. Crossref, Web of Science, Google Scholar
8. P. Colella and K. Pao, A projection method for low speed flows, J. Comput. Phys. 149 (1999) 245–269. Crossref, Web of Science, Google Scholar
9. F. Dauvergne, J.-M. Ghidaglia, F. Pascal and J.-M. Rovarch, Renormalization of the numerical diffusion for an upwind finite volume method. Application to the simulation of Kelvin–Helmholtz instability, in Proc. 5th Int. Symp. Finite Volumes for Complex Applications, eds. R. Eymard and J.-M. Hérard (Wiley, 2008), pp. 321–328. Google Scholar
10. P. Degond, S. Jin and J. G. Liu, Mach number uniform asymptotic-preserving gauge schemes for compressible flows, Bull. Inst. Math. Acad. Sinica (N.S.) 2 (2007) 851–892. Google Scholar
11. S. Dellacherie, Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number, J. Comput. Phys. 229 (2010) 978–1016. Crossref, Web of Science, Google Scholar
12. S. Dellacherie, On a low Mach nuclear core model, ESAIM Proc. 35 (2012) 79–106. Crossref, Google Scholar
13. S. Dellacherie, K. Herilus, J. Jung, A. Mekkas and A. Talpaert, Simulation numérique de systèmes de loi de conservation dans un environement SALOME/C $+ +$ via la bibliothèque CDMATH, in preparation. Google Scholar
14. S. Dellacherie, J. Jung and P. Omnes, A low Mach correction for the Godunov scheme applied to the linear wave equation with porosity, in preparation. Google Scholar
15. S. Dellacherie, P. Omnes and F. Rieper, The influence of cell geometry on the Godunov scheme applied to the linear wave equation, J. Comput. Phys. 229 (2010) 5315–5338. Crossref, Web of Science, Google Scholar
16. F. Fillion, A. Chanoine, S. Dellacherie and A. Kumbaro, FLICA-OVAP: A new platform for core thermal-hydraulic studies, Nucl. Eng. and Design 241 (2011) 4348–4358. Crossref, Web of Science, Google Scholar
17. E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, Vol. 118 (Springer, 1996), pp. 215–220. Crossref, Google Scholar
18. H. Guillard, On the behavior of upwind schemes in the low Mach number limit: IV. P0 approximation on triangular and tetrahedral cells, Comput. Fluids 38 (2009) 1969–1972. Crossref, Web of Science, Google Scholar
19. H. Guillard and A. Murrone, On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes, Comput. Fluids 33(4) (2004) 655–675. Crossref, Web of Science, Google Scholar
20. H. Guillard and A. Murrone, Behavior of upwind scheme in the low Mach number limit: III. Preconditioned dissipation for a five equation two phase model, Comput. Fluids 37 (2008) 1209–1224. Crossref, Web of Science, Google Scholar
21. H. Guillard and C. Viozat, On the behavior of upwind schemes in the low Mach number limit, Comput. Fluids 28 (1999) 63–86. Crossref, Web of Science, Google Scholar
22. A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 135 (1997) 260–278. Crossref, Web of Science, Google Scholar
23. P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Commun. Pure Appl. Math. 7 (1954) 159–193. Crossref, Web of Science, Google Scholar
24. P. D. Lax and X.-D. Liu, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput. 19 (1998) 319–340. Crossref, Web of Science, Google Scholar
25. X.-S. Li and C.-W. Gu, An all-speed Roe-type scheme and its asymptotic analysis of low Mach number behaviour, J. Comput. Phys. 227 (2008) 5144–5159. Crossref, Web of Science, Google Scholar
26. X.-S. Li, C.-W. Gu and J.-Z. Xu, Development of Roe-type scheme for all-speed flows based on preconditioning method, Comput. Fluids 38 (2009) 810–817. Crossref, Web of Science, Google Scholar
27. M.-S. Liou, A sequel to Ausm: Ausm $^{+}$ , Part II: Ausm $^{+}$ -up for all speeds, J. Comput. Phys. 214 (2006) 137–170. Crossref, Web of Science, Google Scholar
28. I. Mary and P. Sagaut, Large eddy simulation of flow around an airfoil near stall, AIAA J. 40 (2002) 1139–1145. Crossref, Google Scholar
29. A. Mekkas, A. Talpaert and S. Dellacherie, Toolbox CDMATH (2016), https://github.com/PROJECT-CDMATH/CDMATH. Google Scholar
30. Y. Moguen, S. Dellacherie, P. Bruel and E. Dick, Momentum interpolation for quasi one-dimensional unsteady low Mach number flows with acoustics, in Proc. 11th World Congr. Computational Mechanics (WCCM XI), ECCOMAS, 2014 (2014). Google Scholar
31. K. Oßwald, A. Siegmund, P. Birken, V. Hannemann and A. Meister, L²Roe: A low dissipation version of Roe’s approximate Riemann solver for low Mach numbers, Int. J. Numer. Methods Fluids 81 (2015) 71–86. Crossref, Web of Science, Google Scholar
32. H. Paillère, C. Viozat, A. Kumbaro and I. Toumi, Comparison of low Mach number models for natural convection problems, Heat Mass Transf. 36 (2000) 567–573. Crossref, Web of Science, Google Scholar
33. F. Rieper, Influence of cell geometry on the behaviour of the first-order Roe scheme in the low Mach number regime, in Finite Volumes for Complex Applications V, eds. R. Eymard and J.-M. Hérard (Wiley, 2008), pp. 625–632. Google Scholar
34. F. Rieper, A low Mach number fix for Roe’s approximate Riemann solver, J. Comput. Phys. 230 (2011) 5263–5287. Crossref, Web of Science, Google Scholar
35. F. Rieper and G. Bader, The influence of cell geometry on the accuracy of upwind schemes in the low Mach number regime, J. Comput. Phys. 228 (2009) 2918–2933. Crossref, Web of Science, Google Scholar
36. P. L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys. 43 (1981) 357–372. Crossref, Web of Science, Google Scholar
37. P. L. Roe, Sonic flux formulae, SIAM J. Sci. Statist. Comput. 13 (1992) 611–630. Crossref, Web of Science, Google Scholar
38. V. V. Rusanov, Calculation of intersection of non-steady shock waves with obstacles, Comput. Math. Phys. USSR 1 (1961) 267–279. Google Scholar
39. M. Sabanca, G. Brenner and N. Alemdaroǧlu, Improvements to compressible Euler methods for low-Mach number flows, Int. J. Numer. Methods Fluids 34 (2000) 167–185. Crossref, Web of Science, Google Scholar
40. S. Schochet, Fast singular limits of hyperbolic PDES, J. Differential Equations 114 (1994) 476–512. Crossref, Web of Science, Google Scholar
41. N. Seguin, Modélisation et Simulation Numérique des Écoulements Diphasiques, Ph.D. thesis, Université de Provence-Aix-Marseille I (2002). Google Scholar
42. G. A. Sod, Numerical Methods in Fluid Dynamics. Initial and Initial-Boundary Value Problems, Vol. 1 (Cambridge Univ. Press, 1985). Crossref, Google Scholar
43. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (Springer Science and Business Media, 2013). Google Scholar
44. B. Van Leer, W.-T. Lee and K. G. Powell, Sonic-point capturing, in AIAA 9th Computational Fluid Dynamics Conf. (American Institute of Aeronautics and Astronautics, 1989), pp. 176–187. Google Scholar
45. G. Volpe, Performance of compressible flow codes at low Mach number, AIAA J. 31 (1993) 49–56. Crossref, Google Scholar