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A high-order implicit finite element method for integrating the two-fluid magnetohydrodynamic equations in two dimensions. (English) Zbl 1388.76442
Summary: We describe a new method for solving the time-dependent two-fluid magnetohydrodynamic (2F-MHD) equations in two dimensions that has significant advantages over other methods. The stream-function/potential representation of the velocity and magnetic field vectors, while fully general, allows accurate description of nearly incompressible fluid motions and manifestly satisfies the divergence condition on the magnetic field. Through analytic manipulation, the split semi-implicit method breaks the full matrix time advance into four sequential time advances, each involving smaller matrices. The use of a high-order triangular element with continuous first derivatives (\(C^{1}\) continuity) allows the Galerkin method to be applied without introduction of new auxiliary variables (such as the vorticity or the current density). These features, along with the manifestly compact nature of the fully node-based \(C^{1}\) finite elements, lead to minimum size matrices for an unconditionally stable method with order of accuracy \(h^{4}\). The resulting matrices are compatible with direct factorization using SuperLU_dist. We demonstrate the accuracy of the method by presenting examples of two-fluid linear wave propagation, two-fluid linear eigenmodes of a tilting cylinder, and of a challenging nonlinear problem in two-fluid magnetic reconnection.

MSC:
76W05 Magnetohydrodynamics and electrohydrodynamics
76M10 Finite element methods applied to problems in fluid mechanics
Software:
M3D-C; SEL/HiFi; SuperLU
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References:
[1] Gruber, R.; Rappaz, J., Finite element methods in linear ideal MHD, (1985), Springer · Zbl 0573.76001
[2] Schnack, D.; Barnes, D.C.; Brennan, D.P., Computational modeling of fully ionized magnetized plasmas using the fluid approximation, Phys. plasma, 13, (2006), (Art. No. 058103)
[3] Jardin, S.C., A triangular finite element with first-derivative continuity applied to fusion MHD applications, J. comput. phys, 200, 133, (2004) · Zbl 1288.76043
[4] Jardin, S.C.; Breslau, J.A., Implicit solution of the four-field extended-magnetohydrodynamic equations using high-order high-continuity finite elements, Phys. plasma, 12, 056101, (2005)
[5] Sovinec, C.R.; Glasser, A.H.; Gianakon, G.A., J. comput. phys., 195, 355, (2004)
[6] Chacon, L.; Knoll, D.A.; Finn, J.M., Implicit, nonlinear reduced resistive MHD nonlinear solver, J. comput. phys, 178, 1, 15-36, (2002) · Zbl 1139.76328
[7] Chacon, L.; Knoll, D.A., A 2D high-beta Hall MHD implicit nonlinear solver, J. comput. phys, 188, 2, 573-592, (2003) · Zbl 1127.76375
[8] Glasser, A.H.; Tang, X.Z., The SEL macroscopic modeling code, Comp. phys. commun., 164, 237, (2004) · Zbl 1196.76006
[9] Strang, G.; Fix, G., An analysis of the finite element method, (1973), Prentice-Hall NJ · Zbl 0278.65116
[10] Zienkiewicz, O.C., The finite element method, (1977), McGraw-Hill London · Zbl 0435.73072
[11] Zlamal, M., Some recent advances in the mathematics of finite elements, (), 59-79 · Zbl 0296.65052
[12] Landau, L.D.; Lifschitz, E.M., Fluid mechanics, (1959), Addison-Wesley Reading, MA · Zbl 0997.70500
[13] Ferraro, N.; Jardin, S.C., Finite element implementation of braginskii’s gyroviscous stress with application to the gravitational instability, Phys. plasmas, 13, 092101, (2006)
[14] Park, W.; Belova, E.V.; Fu, G.Y., Phys. plasmas, 6, 1796-1803, (1999), (Part 2)
[15] Molchanov, I.N.; Nikolenko, L.D., On an approach to integrating boundary problems with a non-unique solution, Inform. process. lett., 1, 168-172, (1972) · Zbl 0243.65059
[16] Richard, R.L.; Sydora, R.D.; Ashour-Abdalla, M., Magnetic reconnection driven by current repulsion, Phys. fluids B, 2, 488, (1990)
[17] Birn, J.; Drake, J.F.; Shay, M.A., J. geophys. res. [space phys.], 106, 3715, (2001)
[18] Breslau, J.; Jardin, S.C., Phys. plasmas, 10, 1291, (2003)
[19] J.W. Demmel, J.R. Gilbert, Y.S. Li, SUPERLU: Users Guide, U.C. Berkeley, 2003.
[20] Braginskii, S.I., Transport processes in a plasma, Reviews of modern physics, vol. 205, (1965), Consultants Bureau New York
[21] Caramana, E.J., J. comput. phys., 96, 484-493, (1991)
[22] Scott, B., Phys. plasmas, 12, 102307, (2005)
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