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Roe matrices for ideal MHD and systematic construction of Roe matrices for systems of conservation laws. (English) Zbl 0919.76053
For ideal MHD an approach for constructing the main Roe scheme’s component, Roe matrix, is presented. First, the authors consider the ideal isentropic MHD in one space dimension. The approach is based on the selection of jumps of flux vector; e.g., the magnetic pressure jump is seeking as sum of jumps of conservative variables with unknown coefficients. These coefficients are defined on the basis of requirement that the resulting Roe matrix retains some properties of the Jacobian of the flux. For ideal MHD in Eulerian and Lagrangian coordinates, the authors construct Roe matrices by using the derived representations for the jumps. For quite general systems of conservation laws, a relationship is established between standard Roe matrix of the same system written in Eulerian and Lagrangian coordinates. The obtained results enable to construct different Roe matrices, if one of them is known. The developed approach is applied to ideal gas dynamics and ideal MHD. Numerical results of calculation for Sod’s like shock tube problem are given.

76M20 Finite difference methods applied to problems in fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
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[1] Daı̈, W.; Woodward, P.R., J. comput. phys., 111, 354, (1994)
[2] Daı̈, W.; Woodward, P.R., J. comput. phys., 115, 485, (1994)
[3] Zachary, A.; Colella, P., J. comput. phys., 99, 341, (1992)
[4] Zachary, A.; Malagoli, A.; Colella, P., SIAM J. sci. comput., 15, 263, (1994)
[5] Woodward, P.R.; Colella, P., Lecture notes in physics, 141, (1981), Springer-Verlag New York/Berlin, p. 434-
[6] Woodward, P.R.; Colella, P., J. comput. phys., 54, 115, (1984)
[7] Woodward, P.R.; Colella, P., J. comput. phys., 54, 174, (1984)
[8] Engquist, B.; Osher, S., Math. comput., 36, 321, (1981)
[9] R. Khanfir, 1995, Approximation volumes finis de type cinétique du système de la MHD idéale à pression isotrope
[10] Powell, K.G., ICASE report, 94-24, (1994)
[11] Brio, M.; Wu, C.C., J. comput. phys., 75, 400, (1988)
[12] Munz, C.D., SIAM J. numer. anal., 31, 17, (1994)
[13] Jeffrey, A.; Taniuti, T., Non-linear wave propagation, (1964), Academic Press New York/London · Zbl 0117.21103
[14] Godunov, S., Résolution numérique des problèmes multidimensionnels de la dynamique des gaz, (1979), Mir Moscow
[15] Roe, P.L., J. comput. phys., 43, 357, (1981)
[16] Godlewski, E.; Raviart, P.A., Collect. ellipses, math. appl., (1991)
[17] Van Kampen, N.G.; Felderhof, B.U., Theoretical methods in plasma physics, (1967), North Holland Amsterdam · Zbl 0159.29601
[18] Lax, P.D., Comm. pure appl. math., 7, 159, (1954)
[19] Aslan, N., Int. J. numer. methods fluids, 23, 1211, (1996)
[20] Roe, P.L.; Balsara, D.S., SIAM J. appl. math., 56, 57, (1996)
[21] Cargo, P.; Gallice, G., C. R. acad. sci. (Paris), 320, 1269, (1995)
[22] Cargo, P.; Gallice, G.; Raviart, P.A., C. R. acad. sci. (Paris), 323, 951, (1996)
[23] Gallice, G., C. R. acad. sci. (Paris), 321, 1069, (1995)
[24] S. Brassier, G. Gallice, 1996, 28th Congrès d’Analyse Numérique, France, 1996
[25] G. Gallice, 1996, Workshop Méthodes Numériques pour la MHD (Paris)
[26] G. Gallice, 1997, Système d’Euler-Poisson, Magnétohydrodynamique et schémas de Roe
[27] Abarbanel, S.; Zwas, G., Math. comput., 23, 549, (1969)
[28] Yee, H.C., Lecture series V.K.I., (1989)
[29] Glaister, P., J. comput. phys., 74, 382, (1988)
[30] Liou, M.S.; van Leer, B.; Shuen, J.S., J. comput. phys., 87, 1, (1990)
[31] Vinokur, M.; Montagne, J.L., J. comput. phys., 89, 276, (1990)
[32] Y. Liu, M. Vinokur, 1989, AIAA Paper, 89-0201
[33] Harten, A.; Lax, P.D.; Van Leer, B., ICASE report, 82-5, (1982)
[34] Harten, A.; Hyman, J.M., J. comput. phys., 50, 235, (1983)
[35] Ryu, D.; Jones, T.W., The astrophysical journal, 442, 228, (1995)
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