Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Numerical approximations of the 10-moment Gaussian closure


Author: Christophe Berthon
Journal: Math. Comp. 75 (2006), 1809-1831
MSC (2000): Primary 65M99, 76N15; Secondary 76P05
DOI: https://doi.org/10.1090/S0025-5718-06-01860-6
Published electronically: June 6, 2006
MathSciNet review: 2240636
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We propose a numerical scheme to approximate the weak solutions of the 10-moment Gaussian closure. The moment Gaussian closure for gas dynamics is governed by a conservative hyperbolic system supplemented by entropy inequalities whose solutions satisfy positiveness of density and tensorial pressure. We consider a Suliciu-type relaxation numerical scheme to approximate the solutions. These methods are proved to satisfy all the expected positiveness properties and all the discrete entropy inequalities. The scheme is illustrated by several numerical experiments.


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

  • 1. D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal., 61 (1996), Nos. 1-2, 163-190. MR 1625520
  • 2. M. Baudin, C. Berthon, F. Coquel, R. Masson, and Q. H. Tran, A relaxation method for two-phase flow models with hydrodynamic closure law, Num. Math., 99 (2005), No.3, 411-440. MR 2117734 (2005h:76079)
  • 3. C. Berthon, Inégalités d'entropie pour un schéma de relaxation, C. R. Acad. Sci. Paris, 340 (2005), No.1, 63-68. MR 2112043
  • 4. C. Berthon, B. Braconnier, and B. Nkonga, Numerical approximation of a degenerated non conservative multifluid model: relaxation scheme, Int. J. Numer. Methods Fluids, 48 (2005), No.1, 85-90.
  • 5. C. Berthon, M. Breuss, and M.-O. Titeux, A relaxation scheme for the approximation of the pressureless Euler equations, Numer. Methods Partial Diff. Eq., 22 (2006), 484-505.
  • 6. C. Berthon and B. Dubroca, work in preparation.
  • 7. F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models, Numer. Math., 94 (2003), No. 4, 623-672. MR 1990588 (2005e:65129)
  • 8. C. Chalons and F. Coquel, Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes, preprint.
  • 9. G.Q. Chen, C.D. Levermore, and T.P. Liu, Hyperbolic Conservation Laws with Stiff Relaxation Terms and Entropy, Comm. Pure Appl. Math., 47 (1995), 787-830. MR 1280989 (95h:35133)
  • 10. F. Coquel, E. Godlewski, A. In, B. Perthame, and P. Rascle, Some new Godunov and relaxation methods for two phase flows, Proceedings of an International Conference on Godunov Methods: Theory and Applications, Kluwer Academic/Plenum Publishers, 2002. MR 1963591 (2004a:76093)
  • 11. F. Coquel and B. Perthame, Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics, SIAM J. Numer. Anal., 35 (1998), 6, 2223-2249. MR 1655844 (2000a:76129)
  • 12. B. Dubroca, M. Tchong, P. Charrier, V.T. Tikhonchuk, and P.-M. Morreeuw, Magnetic field generation in plasma due to anisotropic laser heating, J. Phys. Plasma, accepted.
  • 13. T. I. Gombosi, C. P. T. Groth, P. L. Roe, and S. L. Brown, 35-Moment closure for rarefied gases: Derivation, transport equations, and wave structure, preprint.
  • 14. E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic System of Conservation Laws, Applied Mathematical Sciences, 118, Springer, New York, 1996. MR 1410987 (98d:65109)
  • 15. H. Grad, On the kinetic theory of rarefied gas, Comm. Pure Appl. Math., 2 (1949), 331-407. MR 0033674 (11:473a)
  • 16. A. Harten, P.D. Lax, and B. van Leer, On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws, SIAM Review, 25 (1983), 35-61. MR 0693713 (85h:65188)
  • 17. S. Jin and Z. Xin, The Relaxation Scheme for Systems of Conservation Laws in Arbitrary Space Dimension, Comm. Pure Appl. Math., 45 (1995), 235-276.MR 1322811 (96c:65134)
  • 18. R. J. LeVeque, Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002.MR 1925043 (2003h:65001)
  • 19. C.D. Levermore, Moment closure hierarchies for kinetic theory, J. Statist. Phys., 83 (1996), 1021-1065. MR 1392419 (97e:82041)
  • 20. C.D. Levermore and W.J. Morokoff, The Gaussian moment closure for gas dynamics, SIAM J. Appl. Math., 59 (1998), 1, 72-96. MR 1641154 (99f:76114)
  • 21. R. Liska and B. Wendroff, Comparison of several difference schemes on 1D and 2D test problems for the Euler equations, SIAM J. Sci. Comput., 25 (2003), No.3, 995-1017. MR 2046122 (2004k:76088)
  • 22. T.P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987), 153-175. MR 0872145 (88f:35092)
  • 23. R. Natalini, Convergence to equilibrium for the relaxation approximation of conservation laws, Comm. Pure Appl. Math., 49 (1996), 1-30.MR 1391756 (97c:35131)
  • 24. W.F. Noh, Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, J. Comput. Phys., 72 (1987), 78-120.
  • 25. I. Suliciu, Energy estimates in rate-type thermo-viscoplasticity, Int. J. Plast., 14 (1998), 227-244.
  • 26. J. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. MR 0483954 (58:3905)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M99, 76N15, 76P05

Retrieve articles in all journals with MSC (2000): 65M99, 76N15, 76P05


Additional Information

Christophe Berthon
Affiliation: MAB, UMR 5466 CNRS, Université Bordeaux 1, 351 cours de la libération, 33400 Talence, France
Email: Christophe.Berthon@math.u-bordeaux1.fr

DOI: https://doi.org/10.1090/S0025-5718-06-01860-6
Keywords: Hyperbolic system of conservation laws, Gaussian moment closure, relaxation scheme, discrete entropy inequalities
Received by editor(s): August 4, 2004
Received by editor(s) in revised form: September 13, 2005
Published electronically: June 6, 2006
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society