zbMATH — the first resource for mathematics

An implicit, nonlinear reduced resistive MHD solver. (English) Zbl 1139.76328
Summary: Implicit time differencing of the resistive magnetohydrodynamic (MHD) equations can step over the limiting time scales – such as Alfvén time scales – to resolve the dynamic time scales of interest. However, nonlinearities present in these equations make an implicit implementation cumbersome. Here, viable paths for an implicit, nonlinear time integration of the MHD equations are explored using a 2D reduced viscoresistive MHD model. The implicit time integration is performed using the Newton-Raphson iterative algorithm, employing Krylov iterative techniques for the required algebraic matrix inversions, implemented Jacobian-free (i.e., without ever forming and storing the Jacobian matrix). Convergence in Krylov techniques is accelerated by preconditioning the initial problem. A “physics-based” preconditioner, based on a semi-implicit approximation to the original set of partial differential equations, is employed. The preconditioner employs low-complexity multigrid techniques to invert approximately the resulting elliptic algebraic systems. The resulting 2D reduced resistive MHD implicit algorithm is shown to be successful in dealing with large time steps (on the order of the dynamical time scale of the problem) and fine grids. The algorithm is second-order accurate in time and scalable under grid refinement. Comparison of the implicit CPU time with an explicit integration method demonstrates CPU savings even for moderate (\(64\times 64\)) grids, and close to an order of magnitude in fine grids (\(256\times 256\)).

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76W05 Magnetohydrodynamics and electrohydrodynamics
Full Text: DOI
[1] Lindemuth, I; Killeen, J, Alternating direction implicit techniques for two-dimensional magnetohydrodynamic calculations, J. comput. phys., 13, 181, (1973) · Zbl 0273.76073
[2] Schnack, D; Killeen, J, Nonlinear, two-dimensional magnetohydrodynamic calculations, J. comput. phys., 35, 110, (1980) · Zbl 0428.76106
[3] Harned, D.S; Kerner, W, Semi-implicit method for three-dimensional compressible magnetohydrodynamic simulation, J. comput. phys., 60, 62, (1985) · Zbl 0581.76057
[4] Harned, D.S; Schnack, D.D, Semi-implicit method for long time scale magnetohydrodynamic computations in three dimensions, J. comput. phys., 65, 57, (1986) · Zbl 0591.76187
[5] Schnack, D.D; Barnes, D.C; Harned, D.S; Caramana, E.J, Semi-implicit magnetohydrodynamic calculations, J. comput. phys., 70, 330, (1987) · Zbl 0615.76109
[6] Harned, D.S; Mikic, Z, Accurate semi-implicit treatment of the Hall effect in magnetohydrodynamic computations, J. comput. phys., 83, 1, (1989) · Zbl 0672.76051
[7] Strikwerda, J.C, finite difference schemes and partial differential equations, (1989), Chapman & Hall London/New York · Zbl 0681.65064
[8] Caramana, E.J, Derivation of implicit difference schemes by the method of differential approximation, J. comput. phys., 96, 484, (1991) · Zbl 0732.65082
[9] Jones, O.S; Shumlak, U; Eberhardt, D.S, An implicit scheme for nonideal magnetohydrodynamics, J. comput. phys., 130, 231, (1997) · Zbl 0871.76064
[10] Hujeirat, A, IRMHD: an implicit radiative and magnetohydrodynamical solver for self-gravitating systems, Mon. not. R. astron. soc., 298, 310, (1998)
[11] Hujeirat, A; Rannacher, R, On the efficiency and robustness of implicit methods in computational astrophysics, New astron. rev., 45, 425, (2001)
[12] Hujeirat, A; Rannacher, R, A method for computing compressible, highly stratified flows in astrophysics based on operator splitting, Int. J. numer. methods fluids, 28, 1, (1998) · Zbl 0927.76059
[13] Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing, Boston, 1996. · Zbl 1031.65047
[14] Chan, T.F; Jackson, K.R, Nonlinearly preconditioned Krylov subspace methods for discrete Newton algorithms, SIAM J. sci. stat. comput., 5, 533, (1984) · Zbl 0574.65043
[15] Brown, P.N; Saad, Y, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. sci. stat. comput., 11, 450, (1990) · Zbl 0708.65049
[16] Saad, Y; Schultz, M, GMRES: A generalized minimal residual algorithm for solving non-symetric linear systems, SIAM J. sci. stat. comput., 7, 856, (1986) · Zbl 0599.65018
[17] Mousseau, V.A; Knoll, D.A; Rider, W.J, Physics-based preconditioning and the newton – krylov method for non-equilibrium radiation diffusion, J. comput. phys., 160, 743, (2000) · Zbl 0949.65092
[18] Knoll, D.A; Vanderheyden, W.B; Mousseau, V.A; Kothe, D.B, On preconditioning newton – krylov methods in solidifying flow applications, SIAM J. sci. comput., 23, 381, (2001) · Zbl 1125.65315
[19] Knoll, D.A; Rider, W.J, A multigrid preconditioned newton – krylov method, SIAM J. sci. comput., 21, 691, (1999) · Zbl 0952.65102
[20] Knoll, D.A; Lapenta, G; Brackbill, J.U, A multilevel iterative field solver for implicit, kinetic, plasma simulation, J. comput. phys., 149, 377, (1999) · Zbl 0934.76048
[21] Chacón, L; Barnes, D.C; Knoll, D.A; Miley, G.H, An implicit energy-conservative 2D Fokker-Planck algorithm. II. Jacobian-free Newton-Krylov solver, J. comput. phys., 157, 654, (2000) · Zbl 0961.76058
[22] Strauss, H.R, Nonlinear, 3-dimensional magnetohydrodynamics of noncircular tokamaks, Phys. fluids, 19, 134, (1976)
[23] Drake, J.F; Antonsen, T.M, Nonlinear reduced fluid equations for toroidal plasmas, Phys. fluids, 27, 898, (1984) · Zbl 0555.76097
[24] Hazeltine, R.D; Kotschenreuther, M; Morrison, P.J, A four-field model for tokamak plasma dynamics, Phys. fluids, 28, 2466, (1985) · Zbl 0584.76124
[25] Luskin, M; Rannacher, R, On the smoothing property of the Crank-Nicolson scheme, Appl. anal., 14, 117, (1982) · Zbl 0476.65062
[26] Rannacher, R, Finite element solution of diffusion problems with irregular data, Numer. math., 43, 309, (1984) · Zbl 0512.65082
[27] Leonard, B.P, A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comput. methods appl. mech. eng., 19, 59, (1979) · Zbl 0423.76070
[28] McHugh, P.R; Knoll, D.A, Inexact Newton’s method solution to the incompressible Navier-Stokes and energy equations using standard and matrix-free implementations, Aiaa j., 32, 2394, (1994) · Zbl 0832.76071
[29] Knoll, D.A; McHugh, P.R, Enhanced nonlinear iterative techniques applied to a nonequilibrium plasma flow, SIAM J. sci. comput., 19, 291, (1998) · Zbl 0913.76067
[30] Dembo, R; Eisenstat, S; Steihaug, R, Inexact Newton methods, J. numer. anal., 19, 400, (1982) · Zbl 0478.65030
[31] Kelley, C.T, iterative methods for linear and nonlinear equations, (1995), Soc. for Industr. & Appl. Math Philadelphia · Zbl 0832.65046
[32] Knoll, D.A; Mousseau, V, On Newton-Krylov multigrid methods for the incompressible Navier-Stokes equations, J. comput. phys., 163, 262, (2000) · Zbl 0994.76055
[33] Briggs, W.L, A multigrid tutorial, (1987), Soc. for Industr. & Appl. Math Philadelphia
[34] Harlow, F.H; Amsden, A.A, A numerical fluid dynamical calculation method for all flow speeds, J. comput. phys., 8, 197, (1971) · Zbl 0221.76011
[35] D. A. Knoll, and, L. Chacón, Magnetic reconnection in the two-dimensional Kelvin-Helmholtz instability, Phys. Rev. Lett, to appear.
[36] Knoll, D.A, An improved convection scheme applied to recombining divertor plasma flows, J. comput. phys., 142, 473, (1998) · Zbl 0932.76044
[37] Furth, H.P; Killeen, J; Rosenbluth, M.N, Finite-resistivity instabilities of a sheet pinch, Phys. fluids, 6, 459, (1963)
[38] Kerkhoven, T; Saad, Y, On acceleration methods for coupled nonlinear elliptic systems, Numer. math., 60, 525, (1992) · Zbl 0724.65095
[39] Tannehill, J.C; Anderson, D.A; Pletcher, R.H, computational fluid mechanics and heat transfer, (1997), Taylor & Francis London
[40] Van Leer, B, Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection, J. comput. phys., 23, 276, (1977) · Zbl 0339.76056
[41] Pernice, M; Tocci, M.D, A multigrid-preconditioned Newton-Krylov method for the incompressible Navier-Stokes equations, SIAM J. sci. comput., 23, 398, (2001) · Zbl 0995.76061
[42] M. Pernice, and, R. Hornung, A nonlinear solvers package for SAMRAI, 2002. Available at, http://www.c3.lanl.gov/ pernice/samrai/docs/nlsolvers/html/index.html.
[43] R. Hornung, SAMRAI: Structured adaptive mesh refinement application infrastructure, 2001. Available at, http://www.llnl.gov/CASC/SAMRAI/.
[44] S. Balay, W. D. Gropp, L. C. McInnes, and, B. F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in, Modern Software Tools in Scientific Computing, edited by, E. Arge, A. M. Bruaset, and H. P. Langtangen, Birkhäuser, Basel, 1997, p, 163. · Zbl 0882.65154
[45] S. Balay, W. D. Gropp, L. C. McInnes, and, B. F. Smith, PETSc Users Manual, Technical Report ANL-95/11—Revision 2.1.0, Argonne National Laboratory, 2001.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.