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Why the MUSCL–Hancock Scheme is L1-stable

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Abstract

The finite volume methods are one of the most popular numerical procedure to approximate the weak solutions of hyperbolic systems of conservation laws. They are developed in the framework of first-order numerical schemes. Several approaches are proposed to increase the order of accuracy. The van Leer methods are interesting ways. One of them, namely the MUSCL–Hancock scheme, is full time and space second-order accuracy. In the present work, we exhibit relevant conditions to ensure the L1-stability of the method. A CFL like condition is established, and a suitable limitation procedure for the gradient reconstruction is developed in order to satisfy the stability criterion. In addition, we show that the conservative variables are not useful within the gradient reconstruction and the procedure is extended in the framework of the primitive variables. Numerical experiments are performed to show the interest and the robustness of the method.

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References

  1. Baudin M., Berthon C., Coquel F., Masson R., Tran H. (2005) A relaxation method for two-phase flow models with hydrodynamic closure law. Numer Math 99(3): 411–440

    Article  MATH  MathSciNet  Google Scholar 

  2. Berthon C. (2005) Stability of the MUSCL schemes for the Euler equations. Commun Math Sci 3, 133–158

    MATH  MathSciNet  Google Scholar 

  3. Berthon, C. Numerical approximations of the 10-moment Gaussian closure. Math Comp (accepted)

  4. Bianco F., Puppo G., Russo G. (1999) High-order central schemes for hyperbolic systems of conservation laws. SIAM J Sci Comput 21(1): 294–322

    Article  MATH  MathSciNet  Google Scholar 

  5. Bouchut F. (2004) Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources. Frontiers in Mathematics series, Birkhäuser

    MATH  Google Scholar 

  6. Colella P. (1990) Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 87, 171–200

    Article  MATH  MathSciNet  Google Scholar 

  7. Coquel F., Perthame B. (1998) Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J Numer Anal 35(6): 2223–2249

    Article  MATH  MathSciNet  Google Scholar 

  8. Cordier, S., Buet, C. Asymptotic preserving scheme for radiative hydrodynamics model. (in press)

  9. Durlofsky L.J., Engquist B., Osher S. (1992) Triangle based adaptive stencils for the solution of hyperbolic conservation laws. J Comput Phys 98, 64–73

    Article  MATH  Google Scholar 

  10. Godlewski, E., Raviart, P.A. Hyperbolic systems of conservations laws. SMAI (eds), Ellipse 1991

  11. Godlewski E., Raviart P.A. (1995) Hyperbolic systems of conservations laws. Applied Mathematical Sciences, vol. 118. Springer, Berlin Heidelberg New York

    Google Scholar 

  12. Harten A., Lax P.D., Van Leer B. (1983) On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev 25(1): 35–61

    Article  MATH  MathSciNet  Google Scholar 

  13. Jin S., Xin Z. (1995) The relaxation scheme for systems of conservation laws in arbitrary space dimension. Commun Pure Appl Math 45, 235–276

    Article  MathSciNet  Google Scholar 

  14. Karni S. (1994) A multicomponent flow calculations by a consistent primitive algorithm. J Comp Phys 112, 31–43

    Article  MATH  MathSciNet  Google Scholar 

  15. Khobalatte B., Perthame B. (1994) Maximum principle on the entropy and second-order kinetic schemes. Math of Comput 62(205): 119–131

    Article  MATH  MathSciNet  Google Scholar 

  16. van Leer B. (1979) Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J Comput Phys 32, 101–136

    Article  Google Scholar 

  17. van Leer B. (1984) On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe. SIAM J Sci Statist Comput. 5(1): 1–20

    Article  MATH  MathSciNet  Google Scholar 

  18. LeVeque, R.J, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge Cambridge University Press, (2002)

  19. LiSkA R., Wendroff B. (2003) Comparison of several difference schemes on 1D and 2D test problems for the Euler equations. SIAM J Sci. Comput 25(3): 995–1017

    Article  MATH  MathSciNet  Google Scholar 

  20. Nessyahu H., Tadmor E. (1990) Nonoscillatory central differencing for hyperbolic conservation laws. J Comput Phys 87(2): 408–463

    Article  MATH  MathSciNet  Google Scholar 

  21. Noh W.F. (1987) Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux. J Comput Phys 72, 78–120

    Article  MATH  Google Scholar 

  22. Perthame B., Qiu Y. (1994) A variant of Van Leer’s method for multidimensional systems of conservation laws. J Comput Phys 112(2): 370–381

    Article  MATH  MathSciNet  Google Scholar 

  23. Sanders R., Weiser A. (1992) High resolution staggered mesh approach for nonlinear hyperbolic systems of conservation laws. J Comput Phys 101, 314–329

    Article  MATH  MathSciNet  Google Scholar 

  24. Toro E.F. (1999). Riemann solvers and numerical methods for fluid dynamics. A practical introduction, 2nd edn. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

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Authors and Affiliations

  1. MAB, UMR 5466, Université Bordeaux I, 351 cours de la libération, 33400, Talence, France

    Christophe Berthon

  2. NRIA Futurs, projet ScAlApplix, Domaine de Voluceau-Rocquencourt, B.P. 105, 78153, Le Chesnay Cedex, France

    Christophe Berthon

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  1. Christophe Berthon
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Correspondence to Christophe Berthon.

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Berthon, C. Why the MUSCL–Hancock Scheme is L1-stable. Numer. Math. 104, 27–46 (2006). https://doi.org/10.1007/s00211-006-0007-4

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  • DOI: https://doi.org/10.1007/s00211-006-0007-4

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