A positive MUSCL-Hancock scheme for ideal magnetohydrodynamics

Journal of Computational Physics

Volumn 228, Issue 23, 2009, Pages 8609-8626

  Waagan, K ^a  

^a NATIONAL CENTER FOR ATMOSPHERIC RESEARCH   (United States)

Author keywords

Finite volume methods; Magnetohydrodynamics; Positive conservative schemes

Indexed keywords

MACH NUMBER; MAGNETOHYDRODYNAMICS; MAGNETOPLASMA; STABILITY;

APPROXIMATE RIEMANN SOLVER; CONSERVATIVE SCHEMES; EULER'S EQUATION; FINITE-VOLUME METHOD; IDEAL MAGNETOHYDRODYNAMICS; MASS DENSITIES; MHD EQUATIONS; POSITIVE CONSERVATIVE SCHEME; PREDICTOR CORRECTOR; SECOND ORDERS;

FINITE VOLUME METHOD;

EID: 70349750479     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2009.08.020     Document Type: Article

References (28)

1. Audusse E., Bouchut F., Bristeau M.-O., Klein R., and Perthame B. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 6 (2004) 2050-2065
2. Balsara D.S., and Spicer D.S. A staggered mesh algorithm using high order Godunov fluxes to ensure solenodial magnetic fields in magnetohydrodynamic simulations. J. Comput. Phys. 149 2 (1999) 270-292
3. Berthon C. Stability of the MUSCL schemes for the Euler equations. Commun. Math. Sci. 3 2 (2005) 133-157
4. Berthon C. Why the MUSCL-hancock scheme is L1-stable. Numer. Math. 104 1 (2006) 27-46
5. Bouchut F., Bourdarias Ch., and Perthame B. A MUSCL method satisfying all the numerical entropy inequalities. Math. Comput. 65 216 (1996) 1439-1461
6. Bouchut F. Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. 94 4 (2003) 623-672
7. Bouchut F. Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics (2004), Birkhäuser, Basel viii, p. 135
8. Bouchut F., Klingenberg C., and Waagan K. A multiwave approximate Riemann solver for ideal MHD based on relaxation I - Theoretical framework. Numer. Mathe. 108 1 (2007) 7-41
9. François Bouchut, Christian Klingenberg, Knut Waagan, A multiwave approximate Riemann solver for ideal MHD based on relaxation II - numerical implementation with 3 and 5 waves, submitted for publication. URL: .
10. Brackbill J.U., and Barnes D.C. The effect of nonzero product of magnetic gradient and B on the numerical solution of the magnetohydrodynamic equations. J. Computat. Phys. 35 (1980) 426-430 May
11. Brio M., and Wu C.C. An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 75 2 (1988) 400-422
12. Colella P., and Woodward P.R. The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54 (1984) 174-201
13. Coquel F., and LeFloch P.G. A second order entropy satisfying scheme for systems of conservation laws (Un schéma entropique du second ordre pour les systèmes de lois de conservation.). C.R. Acad. Sci., Paris, Sér. I 320 10 (1995) 1263-1268
14. Dedner A., Kemm F., Kröner D., Munz C.-D., Schnitzer T., and Wesenberg M. Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175 2 (2002) 645-673
15. Fryxell B., Olson K., Ricker P., Timmes F.X., Zingale M., Lamb D.Q., MacNeice P., Rosner R., Truran J.W., and Tufo H. Flash: An adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes. Astrophys. J. Suppl. Ser. 131 1 (2000) 273-334
16. F. Fuchs, A. McMurry, S. Mishra, N.H. Risebro, K. Waagan, Approximate Riemann-solver based high-order finite volume schemes for the Godunov-Powell form of the ideal MHD equations in multi-dimensions, submitted for publication.
17. Gurski K.F. An HLLC-type approximate Riemann solver for ideal magnetohydrodynamics. SIAM J. Sci. Comput. 25 6 (2004) 2165-2187
18. Harten A., Lax P.D., and van Leer B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35-61
19. LeVeque R.J. Numerical Methods for Conservation Laws. second ed. (1992), Birkhäuser, Basel, Switzerland, Boston, USA
20. Li S. An HLLC Riemann solver for magneto-hydrodynamics. J. Comput. Phys. 203 1 (2005) 344-357
21. T.J. Linde, A three-dimensional adaptive multifluid MHD model of the heliosphere, Ph.D. Thesis, University of Michigan, 1998.
22. Marder B. A method for incorporating Gauss' law into electromagnetic pic codes. J. Comput. Phys. 68 1 (1987) 48-55
23. Miyoshi T., and Kusano K. A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys. 208 1 (2005) 315-344
24. Perthame B., and Shu C.-W. On positivity preserving finite volume schemes for Euler equations. Numer. Math. 73 1 (1996) 119-130
25. Kenneth G. Powell, 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.
26. Toro E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics. second ed. A Practical Introduction (1999), Springer, Berlin xix, p. 624
27. Tóth G. The ∇ · B = 0 constraint in shock-capturing magnetohydrodynamics codes. J. Computat. Phys. 161 (2000) 605-652 July
28. Leer B.v. On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe. SIAM J. Sci. Statist. Comput. 5 1 (1984) 1-20