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Asymptotic Preserving Scheme for Euler System with Large Friction

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Abstract

We construct a ‘well-balanced’ and ‘asymptotic preserving’ scheme for the approximation of the model problem of gas dynamics equations with gravity and friction. The friction terms we consider are quite general. We interpret our simple Riemann solver in such a way that the expected properties are directly inherited from the properties of the system of PDEs which is approximated.

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References

  1. Berthon, C., Charrier, P., Dubroca, B.: An HLLC scheme to solve the M1 Model of radiative transfer in two space dimensions. J. Sci. Comput. 31(3), 347–389 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bouchut, F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Frontiers in Mathematics. Birkhäuser, Basel (2004)

    MATH  Google Scholar 

  3. Bouchut, F., Ounaissa, H., Perthame, B.: Upwinding of the source term at interfaces for Euler equations with high friction. Comput. Math. Appl. 53(3–4), 361–375 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Buet, C., Cordier, S.: An asymptotic preserving scheme for hydrodynamics radiative transfer models: numerics for radiative transfer. Numer. Math. 108(2), 199–221 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Buet, C., Després, B.: Asymptotic preserving and positive schemes for radiation hydrodynamics. J. Comput. Phys. 215(2), 717–740 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cargo, P., Le Roux, A.-Y.: Un schéma équilibre adapté au modèle d’atmosphère avec termes de gravité. C. R. Acad. Sci. Paris, Ser. I 318, 73–76 (1994)

    MATH  Google Scholar 

  7. Chalons, C., Coquel, F., Godlewski, E., Raviart, P.-A., Seguin, N.: Godunov-type schemes for hyperbolic systems with parameter dependent source. The case of Euler system with friction. Math. Models Methods Appl. Sci. 20(11), 2109–2166 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Coquel, F., Godlewski, E., Perthame, B., In, A., Rascle, P.: Some new Godunov and relaxation methods for two-phase flow problems. In: Godunov Methods, Oxford, 1999, pp. 179–188. Kluwer Academic/Plenum, Dordrecht/New York (2001)

    Google Scholar 

  9. Gallice, G.: Solveurs simples positifs et entropiques pour les systèmes hyperboliques avec terme source. C. R. Math. Acad. Sci. Paris 334(8), 713–716 (2002), (French). [Entropic Godunov-type schemes for hyperbolic systems with source term]

    MATH  MathSciNet  Google Scholar 

  10. Gosse, L., Toscani, G.: Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes. SIAM J. Numer. Anal. 41(2), 641–658 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gosse, L., Toscani, G.: Asymptotic-preserving & well-balanced schemes for radiative transfer and the Rosseland approximation. Numer. Math. 98(2), 223–250 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Greenberg, J.M., Leroux, A.-Y.: A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33(1), 1–16 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hsiao, L., Liu, T.-P.: Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Commun. Math. Phys. 143(3), 599–605 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Huang, F., Marcati, P., Pan, R.: Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum. Arch. Ration. Mech. Anal. 176, 1–24 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jin, S.: Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 122, 51–67 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jin, S.: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21, 441–454 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Jin, S., Levermore, C.D.: Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 126, 449–467 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Klar, A.: An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit. SIAM J. Numer. Anal. 35(3), 1073–1094 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Marcati, P., Milani, A.: The one-dimensional Darcy’s law as the limit of a compressible Euler flow. J. Differ. Equ. 84(1), 129–147 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  20. Naldi, G., Pareschi, L.: Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation. SIAM J. Numer. Anal. 37(4), 1246–1270 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. CNRS and CMAP, Ecole Polytechnique, Paris, France

    F. Coquel

  2. Laboratoire Jacques-Louis Lions, UPMC Univ. Pierre et Marie Curie and CNRS, Paris, France

    E. Godlewski

Authors
  1. F. Coquel
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  2. E. Godlewski
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Correspondence to E. Godlewski.

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Coquel, F., Godlewski, E. Asymptotic Preserving Scheme for Euler System with Large Friction. J Sci Comput 48, 164–172 (2011). https://doi.org/10.1007/s10915-011-9459-5

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  • DOI: https://doi.org/10.1007/s10915-011-9459-5

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