A multiwave approximate Riemann solver for ideal MHD based on relaxation. I: Theoretical framework

Numerische Mathematik

Volumn 108, Issue 1, 2007, Pages 7-42

  Bouchut, François ^a   Klingenberg, Christian ^b   Waagan, Knut ^c  

^a ECOLE NORMALE SUPÉRIEURE   (France)

^b UNIVERSITY OF WÜRZBURG   (Germany)

^c UNIVERSITY OF OSLO   (Norway)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 35548939884     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-007-0108-8     Document Type: Article

References (26)

1. Batten P., Clarke N., Lambert C. and Causon D.M. (1997). On the choice of wavespeeds for the HLLC Riemann solver. SIAM J. Sci. Comput. 18(6): 1553-1570
2. Bezard F. and Després B. (1999). An entropic solver for ideal Lagrangian magnetohydrodynamics. J. Comput. Phys. 154(1): 65-89
3. Bouchut F. (2003). Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. 94(4): 623-672
4. Bouchut, F.: Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics, vol. viii, 135 p. Birkhäuser, Basel (2004)
5. Brenier Y. (1984). Averaged multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal. 21: 1013-1037
6. Brio M. and Wu C.C. (1988). An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 75(2): 400-422
7. Cargo P. and Gallice G. (1997). Roe matrices for ideal MHD and systematic construction of Roe matrices for systems of conservation laws. J. Comput. Phys. 136(2): 446-466
8. Chen G.-Q., Levermore C.D. and Liu T.-P. (1994). Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47(6): 787-830
9. Coquel, F., Godlewski, E., Perthame, B., In, A., Rascle, P.: Some new Godunov and relaxation methods for two-phase flow problems. In: Toro, E.F. (ed.) Godunov methods. Theory and applications, pp. 179-188. International conference, Oxford, GB, October 1999. Kluwer Academic/Plenum Publishers, New York (2001)
10. Davis S.F. (1988). Simplified second-order Godunov-type methods. SIAM J. Sci. Stat. Comput. 9(3): 445-473
11. Dedner A., Kemm F., Krner D., Munz C.-D., Schnitzer T. and Wesenberg M. (2002). Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175(2): 645-673
12. Després B. (2001). Lagrangian systems of conservation laws. Invariance properties of Lagrangian systems of conservation laws, approximate Riemann solvers and the entropy condition. Numer. Math. 89(1): 99-134
13. Einfeldt B., Munz C.D., Roe P.L. and Sjögreen B. (1991). On godunov-type methods near low densities. J. Comput. Phys. 92(2): 273-295
14. Einfeldt B. (1988). On Godunov-type methods for gas dynamics. SIAM J. Numer. Anal. 25(2): 294-318
15. Gallice G. (2003). Positive and entropy stable Godunov-type schemes for gas dynamics and MHD equations in Lagrangian or Eulerian coordinates. Numer. Math. 94(4): 673-713
16. Gurski K.F. (2004). An HLLC-type approximate Riemann solver for ideal magnetohydrodynamics. SIAM J. Sci. Comput. 25(6): 2165-2187
17. Harten A., Lax P.D. and van Leer B. (1983). On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25: 35-61
18. Jin S. and Xin Z. (1995). The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48(3): 235-276
19. Li S. (2005). An HLLC Riemann solver for magneto-hydrodynamics. J. Comput. Phys. 203(1): 344-357
20. Miyoshi T. and Kusano K. (2005). A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys. 208(1): 315-344
21. Powell, K.G.: An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical report, Institute for Computer Applications in Science and Engineering (ICASE) (1994)
22. Tang H. and Xu K. (2000). A high-order gas-kinetic method for multidimensional ideal magnetohydrodynamics. J. Comput. Phys. 165(1): 69-88
23. Toro E.F., Spruce M. and Speares W. (1994). Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4(1): 25-34
24. Toro, E.F.: Riemann solvers and numerical methods for fluid dynamics. A practical introduction, 2nd edn, vol. xix, 624 p. Springer, Berlin (1999)
25. Tzavaras A.E. (1999). Materials with internal variables and relaxation to conservation laws. Arch. Ration. Mech. Anal. 146(2): 129-155
26. Xu K. (1999). Gas-kinetic theory-based flux splitting method for ideal magnetohydrodynamics. J. Comput. Phys. 153(2): 334-352