Quasimonotonous 2D MHD scheme for unstructured meshes

V. A. Gasilov ; S. V. D'yachenko

Matematičeskoe modelirovanie, Tome 17 (2005) no. 12, pp. 87-109.

Résumé

The generalized Lax–Friedrichs scheme is designed for the ideal MHD system. The governing system representing a 2,5D flow model is considered for plane geometry. The second-order difference approximations are done to the MHD equations written in the form of conservation laws. The set of control volumes is conjoined with basic triangular mesh and is generated starting from some initial partition which is usually the Voronoi diagram of basic triangulation. The procedure of time–advance is an explicit predictor-corrector which provides the second order approximation to MHD system in time. The constructed algo-rithms are examined in numerical experiments with known MHD problems related to flows resulted from break of MHD discontinuities and the test results are presented.

Bibliographie

[1] J. P. Boris, D. L. Book, “Flux corrected transport I, SHASTA. A fluid transport algorithm that works”, J. Comp. Phys., 11 (1973), 38–69 | DOI | MR | Zbl

[2] C. R. DeVore, “Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics”, J. Comput. Phys., 92 (1991), 142–160 | DOI | MR | Zbl

[3] M. A. Ryazanov, S. F. Krylov, V. F. Tishkin, K. V. Vyaznikov, “Raznostnye skhemy dlya dvumernykh zadach magnitnoi gidrodinamiki s poloidalnym polem”, Matematicheskoe modelirovanie, 4:10 (1992), 47–61 | MR

[4] K. Sankaran, L. Martinelli, S. C. Jardin, and E. Y. Chueiri, “A flux-limited method for solving the MHD equations to simulate propulsive plasma flows”, Int. J. Numer. Meth. Engng., 53 (2002), 1415–1432 | DOI | MR | Zbl

[5] D. Kroner, Numerical schemes for conservation laws, B. G. Teubner Publ., Stutgart, Leipzig, 2000 | MR

[6] A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical problems of the numerical solution of hyperbolic equations, Fizmatlit Publ. House, M., 2001 | MR

[7] W. Dai, P. R. Woodward, “An approximate Riemann solver for ideal magnetohydrodynamics”, J. Comput. Phys., 111:2 (1994), 354–372 | DOI | MR | Zbl

[8] M. Brio, C. Wu, “An upwind differencing scheme for the equations of ideal magnetohydrodynamics”, J. Comput. Phys., 75:2 (1988), 400–422 | DOI | MR | Zbl

[9] A. L. Zachary, P. Colella, “A higher-order Godunov method for the equations of ideal magnetohydrodynamics”, J. Comput. Phys., 92:2 (1992), 341–347 | DOI | MR | Zbl

[10] K. M. Magomedov, A. S. Kholodov, Setochno-kharakteristicheskie chislennye metody, Nauka, M., 1988 | MR

[11] A. S. Boldarev, V. A. Gasilov, O. G. Olkhovskaya, V. M. Panin, Primenenie raznostnykh skhem s korrektsiei potokov k uravneniyam odnomernoi gazovoi dinamiki i magnitnoi gidrodinamiki, Preprint No 8, IMM RAN, M., 1993 | MR

[12] A. L. Zachary, A. Malagoli, P. Colella, “A higher-order Godunov method for multidimensional ideal magneto-hydrodynamics”, SIAM J. Sci. Comput., 15:2 (1994), 263–284 | DOI | MR | Zbl

[13] T. Tanaka, “Finite volume TVD schemes on an unstructured grid system for three-dimensional MHD simula-tions of inhomogeneous systems including strong background potential fields”, J. Comput. Phys., 111:2 (1994), 381–389 | DOI | MR | Zbl

[14] Alexander Kurganov and Eitan Tadmor, “New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations”, Journ. Comput. Phys., 160:3 (2000), 241–282 | MR | Zbl

[15] Jorge Balbas, Eitan Tadmor and Cheng-Chin Wu, Non-oscillatory central schemes for one- and two-dimensional MHD equations, CSCAMM Report 03-16, Universiy of Maryland, College Park, 2003 | MR

[16] A. E. Dudorov, A. G. Zhilkin, O. A. Kuznetsov, “Quasimonotonous difference schemes of higher accuracy for the equations of magnetohydrodynamics”, Mathematical Modeling, 11:1 (1999), 101–116 | MR

[17] A. E. Dudorov, A. G. Zhilkin, O. A. Kuznetsov, “Two-dimensional numerical code for axially symmetrical and self-gravitational MHD flows”, Mathematical Modeling, 11:11 (1999), 109–127 | MR

[18] V. A. Gasilov, S. V. D'yachenko, E. L. Kartasheva, “Algorithm of control volume mesh generation for two-dimensional domains of complex geometry”, Fundamental Problems of Physics and Mathematics and Simulation of Engineering and Technology Problems, Review of Sci. Publ. 5th Issue, Moscow State University of Technology “STANKIN”, 2003, 131–140 | MR

[19] Franco P. Preparata, Michael Ian Shamos, Computational Geometry: An Introduction, Springer Verlag Publ. House, N.Y., August 1993 | MR

[20] A. A. Samarskii, Yu. P. Popov, Raznostnye metody resheniya zadach gazovoi dinamiki, Nauka, M., 1992 | MR

[21] Van Leer B., “Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second order scheme”, Journ. Comput. Phys., 14 (1974), 361–370 | DOI | MR | Zbl

[22] P. K. Sweby, “High resolution schemes using flux limiters for hyperbolic conservation laws”, SIAM J. Numer. Anal., 21:5 (1984), 995–1011 | DOI | MR | Zbl

[23] K. V. Vyaznikov, V. F. Tishkin, A. P. Favorskii, “Postroenie monotonnykh raznostnykh skhem povyshennogo poryadka approksimatsii dlya sistem uravnenii giperbolicheskogo tipa”, Matem. modelirovanie, 1:5 (1989), 95–120 | MR | Zbl

[24] A. G. Aksenov, Raznostnaya skhema tipa S. K. Godunova dlya integrirovaniya uravnenii MGD, opisyvayuschikh techenie gaza v transversalnom magnitnom pole, Preprint No 19-91-55, ITEF RAN, M., 1991 | MR

[25] S. I. Mukhin, S. V. Popov, Yu. P. Popov, Raznostnye skhemy s iskusstvennoi dispersiei dlya uravnenii magnitogidrodinamiki, Preprint No 19, IMP im. M. V. Keldysha RAN, M., 1985 | MR

[26] V. A. Gasilov, A. S. Chuvatin, A. Yu. Krukovskii, E. L. Kartasheva, O. G. Olkhovskaya, A. S. Boldarev and S. V. D'yachenko, “Magnetically-driven plasma simulations with MARPLE”, Mathematical Modeling: Modern Methods and Applications, The book of scientific articles, ed. Lydmila A. Uvarova, Yanus-K, Moscow, 2004, 126–142 | MR