Abstract
We introduce a flux-splitting formula for the approximation of the ideal MHD equations in conservative form. The Faraday equation is considered as the average of an abstract kinetic equation, giving a flux-splitting formula. For the other part of the equations, we generalize formally the classical half-Maxwellian flux-splitting of the Euler equations. Numerical results on MHD shock tube problems are displayed.
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References
Bouchut, F. (1994). Personal communication.
Bourdel, F., Delorme, P., and Mazet, P. A. (1988). Convexity in hyperbolic problems, inNotes on Num. Fluid Mechanics, Vol. 24, 1988, Ballman and Jeltsch (eds.), Vieweg.
Brio, M., and Wu, C. C. (1988). An upwind differencing scheme for the equations of ideal magnetohydrodynamics,J. Comp. Phys. 75, 400–422.
Coron, F., and Perthame, B. (1991). Numerical passage from kinetic to fluid equations,SIAM Jour. on Num. Anal. 28(1), 26–42.
Croisille, J. P., and Delorme, P. (1992). Kinetic symmetrizations and pressure laws for the Euler equations,Physica D 57, 395–416.
Croisille, J. P., and Villedieu, P. (1992). Kinetic flux-splitting schemes for hypersonic flows,Proc. 13th ICNMFD, Lecture Notes in Physics, Rome, p. 414, Sabetta and Napolitano (eds.).
Deshpande, S. M. (1986a). On the Maxwellian distribution, symmetric form and entropy conservation for the Euler equations, N.A.S.A. T.P. 2583.
Deshpande, S. M. (1986b). A second-order accurate kinetic-theory-based method for inviscid compressible flows, N.A.S.A. T.P. 2613.
Deshpande, S. M. (1986c). Kinetic theory based new upwind methods for inviscid compressible flows, AIAA-86-0275.
Elizarova, T. G., and Chetverushkin, B. N. (1985). Kinetic algorithms for calculating gas dynamics flows,USSR Comput. Math. Phys. 25(5), 164–169.
Harten, A., Lax, P. D., and Van Leer, B. (1983). On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,SIAM Review 25(1), 35–61.
Hirsch, C. (1988).Numerical Computation of Internal and External Flows, Vol. 2 Wiley.
Jeffrey, A., and Taniuti, T. (1964).Nonlinear Wave Propagation, Academic Press, New York.
Kaniel, S. (1979). Approximation of the hydrodynamic equations by a transport process, inApproximation methods for Navier-Stokes problems, Lectures Notes in Math. 771, Rautman, R. (ed.), Springer.
Landau, L., and Lifschitz, E. (1990).Electrodynamique des milieux continus, Mir.
Perthame, B. (1990). Boltzmann type schemes for gas dynamics and the entropy property,SIAM Jour. on Num. Anal. 27(6), 1405–1421.
Perthame, B. (1992). Second order Boltzmann schemes for compressible Euler equations in one and two space dimensions,SIAM. J. Num. Anal. 29(1), 1–19.
Prendergast, K. H., and Xu, K. (1993). Numerical Hydrodynamics from Gas-Kinetic Theory,J. Comput. Phys. 109, 53–66.
Pullin, D. I. (1980). Direct simulation methods for compressible inviscid ideal-gas flow,Jour. of Com. Phys. 34, 231–244.
Sanders, R. H., and Prendergast, K. H. (1974). The possible relation of 3-kiloparsec arm to explosions in the galactic nucleus.The Astrophysical Journal, 188.
Zachary, A. L., and Collela, P. (1992). An higher order Godunov method for the equations of ideal magnetohydrodynamics,J. Comp. Phys. 99, 341–347.
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Croisille, JP., Khanfir, R. & Chanteur, G. Numerical simulation of the MHD equations by a kinetic-type method. J Sci Comput 10, 81–92 (1995). https://doi.org/10.1007/BF02087961
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DOI: https://doi.org/10.1007/BF02087961