Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Nonlinear projection methods for multi-entropies Navier-Stokes systems

Authors: Christophe Berthon and Frédéric Coquel
Journal: Math. Comp. 76 (2007), 1163-1194
MSC (2000): Primary 65M99, 65M12; Secondary 76N15
Published electronically: February 7, 2007
MathSciNet review: 2299770
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is devoted to the numerical approximation of the compressible Navier-Stokes equations with several independent entropies. Various models for complex compressible materials typically enter the proposed framework. The striking novelty over the usual Navier-Stokes equations stems from the generic impossibility of recasting equivalently the present system in full conservation form. Classical finite volume methods are shown to grossly fail in the capture of viscous shock solutions that are of primary interest in the present work. To enforce for validity a set of generalized jump conditions that we introduce, we propose a systematic and effective correction procedure, the so-called nonlinear projection method, and prove that it preserves all the stability properties satisfied by suitable Godunov-type methods. Numerical experiments assess the relevance of the method when exhibiting approximate solutions in close agreement with exact solutions.

References [Enhancements On Off] (What's this?)

  • 1. C. BERTHON, Contributions to the numerical analysis of the compressible Navier-Stokes equations with two specific entropies. Applications to turbulent compressible flows. Ph.D. dissertation (in French) University Paris VI, 1999.
  • 2. C. BERTHON AND F. COQUEL, Travelling wave solutions existence for multi-entropies Navier-Stokes equations, work in preparation, see also Proceedings of the 7th International Conference on Hyperbolic Problems, Zurich, 1998.
  • 3. C. BERTHON AND F. COQUEL, About shock layers for compressible turbulent flow models, work in preparation.
  • 4. J.-F. COLOMBEAU, A. Y. LEROUX, A. NOUSSAÏR AND B. PERROT, Microscopic profiles of shock waves and ambiguities in multiplications of distributions, SIAM J. of Numer. Anal., 26, No 4, 871-883 1989. MR 1005514 (91c:35086)
  • 5. F. COQUEL AND P. LEFLOCH, Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach, Math. Comp., 57, 169-210 1991. MR 1079010 (91m:65229)
  • 6. F. COQUEL AND C. MARMIGNON, A Roe-type linearization for the Euler equations for weakly ionized multi-component and multi-temperature gas, Proceedings of the AIAA 12$ ^{th}$ CFD Conference, San Diego (USA) 1995.
  • 7. F. COQUEL AND B. PERTHAME, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics, SIAM J. of Numer. Anal., 35, No 6, 2223-2249 1998. MR 1655844 (2000a:76129)
  • 8. G. DAL MASO, P. LEFLOCH AND F. MURAT, Definition and weak stability of nonconservative products, J. Math. Pures Appl., 74, 483-548 1995. MR 1365258 (97b:46052)
  • 9. A. FORESTIER, J.-M. H´ERARD AND X. LOUIS, A Godunov-type solver to compute turbulent compressible flows, C.R. Acad. Sci. Paris, 324, Série I, No 8, pp. 919-926 1997. MR 1450449 (98g:76048)
  • 10. D. GILBARG, The existence and limit behavior of the one-dimensional shock layer, Amer. J. Math., 73, 256-274 1951. MR 0044315 (13:401e)
  • 11. E. GODLEWSKI AND P.-A. RAVIART, Hyperbolic systems of conservation laws, Applied Mathematical Sciences, Vol 118, Springer 1996. MR 1410987 (98d:65109)
  • 12. T. Y. HOU AND P. G. LEFLOCH, Why nonconservative schemes converge to wrong solutions: error analysis, Math. of Comp., Vol 62, No 206, 497-530 1994. MR 1201068 (94g:65093)
  • 13. S. KARNI, Viscous shock profiles and primitive formulations, SIAM J. Numer. Anal., 29, No 6, 1592-1609 1992. MR 1191138 (93j:65163)
  • 14. B. LARROUTUROU, How to preserve the mass fractions positivity when computing compressible multi-component flows, J. Comput. Phys. 95, No 1, 59-84 1991. MR 1112315 (92k:76069)
  • 15. B. LARROUTUROU AND C. OLIVIER, On the numerical approximation of the K-eps turbulence model for two dimensional compressible flows, INRIA report, No 1526 1991.
  • 16. P.D. LAX AND B. WENDROFF, Systems of conservation laws, Comm. Pure Appl. Math., Vol 13, 217-237 1960. MR 0120774 (22:11523)
  • 17. P.G. LEFLOCH, Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form. Comm. Part. Diff. Equa. 13, No 6, 669-727 (1988). MR 0934378 (89h:35194)
  • 18. F. R. MENTER, Zonal two equation $ (k-\omega)$ turbulence model for aerodynamic flows, 24th AIAA fluid dynamics conference, Orlando, (1993).
  • 19. F. R. MENTER, Improved two equation $ (k-\omega)$ turbulence model for aerodynamic flows, NASA Technical Report 103975, (1992).
  • 20. B. MOHAMMADI AND O. PIRONNEAU, Analysis of the k-Epsilon Turbulence Model, Research in Applied Mathematics, Masson, Paris, 1994. MR 1296252 (95i:76048)
  • 21. P.-A. RAVIART AND L. SAINSAULIEU, A nonconservative hyperbolic system modeling spray dynamics. Part 1. Solution of the Riemann problem, Math. Models Methods in Appl. Sci., 5, No 3, 297-333 1995. MR 1330136 (96a:76096)
  • 22. P.L. ROE, Approximate Riemann solvers, parameter vectors and difference schemes, J. Comp. Phys., 43, 357-372 1981. MR 0640362 (82k:65055)
  • 23. L. SAINSAULIEU, Traveling waves solutions of convection-diffusion systems whose convection terms are weakly nonconservative, SIAM J. Appl. Math., 55, No 6, 1552-1576 1995. MR 1358789 (96m:65097)
  • 24. B. R. SMITH, A near wall model for the $ (k-l)$ two equation turbulence model, 25th AIAA fluid dynamics conference, Colorado Springs, (1994).
  • 25. E. TADMOR, A minimum entropy principle in the gas dynamics equations, Appl. Numer. Math., No 2, 211-219 1986. MR 0863987 (88a:76037)
  • 26. E. TADMOR, The numerical viscosity of entropy stable schemes for systems of conservation laws, I, Math. Comp., No 49, 91-103 1987. MR 0890255 (88k:65087)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M99, 65M12, 76N15

Retrieve articles in all journals with MSC (2000): 65M99, 65M12, 76N15

Additional Information

Christophe Berthon
Affiliation: MAB, UMR 5466 CNRS, Université Bordeaux I, 351 cours de la libération, 33405 Talence Cedex, France

Frédéric Coquel
Affiliation: CNRS and Laboratoire Jacques-Louis Lions, UMR 7598, Tour 55-65, Université Pierre et Marie Curie, BC 187, 75252 Paris Cedex 05, France.

Keywords: Navier--Stokes equations, entropy inequalities, nonconservative products, travelling wave solutions, Godunov-type methods, discrete entropy inequalities, nonlinear projection, turbulence models
Received by editor(s): March 2, 2005
Received by editor(s) in revised form: April 16, 2006
Published electronically: February 7, 2007
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society