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Implicit symmetrized streamfunction formulations of magnetohydrodynamics. (English) Zbl 1149.76028

Summary: We apply the finite element method to the classic tilt instability problem of two-dimensional, incompressible magnetohydrodynamics, using a streamfunction approach to enforce the divergence-free conditions on magnetic and velocity fields. We compare two formulations of the governing equations, the standard one based on streamfunctions and a hybrid formulation with velocities and magnetic field components. We use a finite element discretization on unstructured meshes and an implicit time discretization scheme. We use the PETSc library with index sets for parallelization. To solve the nonlinear problems on each time step, we compare two nonlinear Gauss-Seidel-type methods and Newton’s method with several time-step sizes. We use GMRES in PETSc with multigrid preconditioning to solve the linear subproblems within the nonlinear solvers. We also study the scalability of this simulation on a cluster.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows

Software:

PETSc; Wesseling
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Full Text: DOI

References:

[1] Gropp, Proceedings of SC2000 (2000)
[2] Mavriplis, An assessment of linear versus non-linear multigrid methods for unstructured mesh solvers, Journal of Computational Physics 175 pp 302– (2002) · Zbl 0995.65099
[3] Strauss, An adaptive finite element method for magnetohydrodynamics, Journal of Computational Physics 147 pp 318– (1998) · Zbl 0936.76032
[4] Lankalapalli, Adaptive finite element analysis and error estimation for magnetohydrodynamics, Journal of Computational Physics (2006) · Zbl 1118.76039
[5] Brenner, The Mathematical Theory of Finite Element Methods (1994) · Zbl 0804.65101
[6] Ciarlet, The Finite Element Method for Elliptic Problems (1988)
[7] Girault, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms (1986) · Zbl 0585.65077
[8] Chacon, An implicit, nonlinear reduced resistive MHD solver, Journal of Computational Physics 178 pp 15– (2002)
[9] Gropp, High performance parallel implicit CFD, Parallel Computing 27 pp 337– (2001) · Zbl 0971.68191
[10] Hujeirat, On the efficiency and robustness of implicit methods in computational astrophysics, New Astronomy Reviews 45 pp 425– (2001)
[11] Jones, An implicit scheme for nonideal magnetohydrodynamics, Journal of Computational Physics 130 pp 231– (1997) · Zbl 0871.76064
[12] Keyes, Contemporary Mathematics 306 pp 29– (2001)
[13] Knoll, A multilevel iterative field solver for implicit, kinetic, plasma simulation, Journal of Computational Physics 149 pp 377– (1999) · Zbl 0934.76048
[14] Turek, A comparison study of time-stepping techniques for the incompressible Navier-Stokes equations: from fully implicit non-linear schemes to semi-implicit projection methods, International Journal for Numerical Methods in Fluids 22 pp 987– (1996) · Zbl 0864.76052
[15] Cai, Nonlinearly preconditioned inexact Newton algorithms, SIAM Journal on Scientific Computing 24 pp 183– (2002) · Zbl 1015.65058
[16] D’Amico, A Newton-Raphson approach for nonlinear diffusion equations in radiation hydrodynamics, Journal of Quantitative Spectroscopy and Radiative Transfer 54 pp 655– (1995)
[17] Eisenstat, Globally convergent inexact Newton methods, SIAM Journal on Optimization 4 pp 393– (1994) · Zbl 0814.65049
[18] Knoll, Jacobian-free Newton-Krylov methods: a survey of approach and applications, Journal of Computational Physics 193 pp 357– (2004)
[19] Knoll, A multigrid preconditioned Newton-Krylov method, SIAM Journal on Scientific Computing 21 pp 691– (1999)
[20] Kaushik, Proceedings of the 11th International Conference on Domain Decomposition Methods pp 513– (1998)
[21] Saad, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing 7 pp 856– (1986) · Zbl 0599.65018
[22] Bramble, Multigrid Methods (1993)
[23] Brandt, Multigrid Techniques: 1984 Guide, with Applications to Fluid Dynamics (1984)
[24] Hackbush, Multigrid Methods and Applications (1985)
[25] Wesseling, An Introduction to Multigrid Methods (1992) · Zbl 0760.65092
[26] Biskamp, Nonlinear Magnetohydrodynamics (1997)
[27] Davidson, An Introduction to Magnetohydrodynamics (2001) · Zbl 0974.76002
[28] Philip, Domain Decomposition Methods in Science and Engineering XVI pp 723– (2006)
[29] Jardin, A triangular finite element with first-derivative continuity applied to fusion MHD applications, Journal of Computational Physics 200 pp 133– (2004) · Zbl 1288.76043
[30] Kang, Domain Decomposition Methods in Science and Engineering XVI pp 619– (2006)
[31] Balay S, Gropp WD, McInnes LC, Smith BF. PETSc Users Manual. Technical Report ANL-95/11-Revision 2.1.3, Argonne National Laboratory, 2001.
[32] Brenan, Numerical Solution of Initial Value Problems in Differential-Algebraic Equations (1995)
[33] Richard, Magnetic reconnection driven by current repulsion, Physics of Fluids B 2 pp 488– (1990)
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