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Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods. (English) Zbl 1425.76312

Summary: This paper explores the development of a scalable, nonlinear, fully-implicit stabilized unstructured finite element (FE) capability for 2D incompressible (reduced) resistive MHD. The discussion considers the implementation of a stabilized FE formulation in context of a fully-implicit time integration and direct-to-steady-state solution capability. The nonlinear solver strategy employs Newton-Krylov methods, which are preconditioned using fully-coupled algebraic multilevel preconditioners. These preconditioners are shown to enable a robust, scalable and efficient solution approach for the large-scale sparse linear systems generated by the Newton linearization. Verification results demonstrate the expected order-of-accuracy for the stabilized FE discretization. The approach is tested on a variety of prototype problems, including both low-Lundquist number (e.g., an MHD Faraday conduction pump and a hydromagnetic Rayleigh-Bernard linear stability calculation) and moderately-high Lundquist number (magnetic island coalescence problem) examples. Initial results that explore the scaling of the solution methods are presented on up to 4096 processors for problems with up to 64M unknowns on a CrayXT3/4. Additionally, a large-scale proof-of-capability calculation for 1 billion unknowns for the MHD Faraday pump problem on 24,000 cores is presented.

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
76M10 Finite element methods applied to problems in fluid mechanics
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[1] Goedbloed, H.; Poedts, S., Principles of Magnetohydrodynamics with Applications to Laboratory and Astrophysical Plasmas (2004), Cambridge University Press
[2] Ascher, U. M.; Petzold, L. R., Computer methods for ordinary differential equations and differential-algebraic equations, SIAM (1998) · Zbl 0908.65055
[3] Dai, W.; Woodward, P. R., A simple finite difference scheme for multi-dimensional magnetohydrodynamic equations, J. Comput. Phys., 142, 331 (1998)
[4] Ryu, D.; Miniati, F.; Jones, T. W.; Frank, A., A divergence-free upwind code for multi-dimensional magnetohydrodynamics flows, Astrophys. J., 509, 244 (1998)
[5] Balsara, D. S., Divergence-free adaptive mesh refinement for magnetohydrodynamics, J. Comput. Phys., 174, 614-648 (2001) · Zbl 1157.76369
[6] Tóth, G., The ∇·\(B\)=0 constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161, 605-652 (2000) · Zbl 0980.76051
[7] Tóth, G.; Keppens, R.; Botchev, M. A., Implicit and semi-implicit schemes in the Versatile Advection Code: numerical tests, Astron. Astrophys., 332, 1159-1170 (1998)
[8] Keppens, R.; Tóth, G.; Botchev, M. A.; Ploeg, A. V.D., Implicit and semi-implicit schemes: algorithms, Int. J. Numer. Meth. Fl., 30, 335-352 (1999) · Zbl 0951.76059
[9] Aydemir, A. Y.; Barnes, D. C., An implicit algorithm for compressible three-dimensional magnetohydrodynamic calculations, J. Comput. Phys., 59, 1, 108-119 (1985) · Zbl 0568.76112
[10] Park, W.; Breslau, J.; Chen, J.; Fu, G. Y.; Jardin, S. C.; Klasky, S.; Menard, J.; Pletzer, A.; Stratton, B. C.; Stutman, D.; Strauss, H. R.; Sugiyama, L. E., Nonlinear simulation studies of Tokamaks and STS, Nucl. Fusion, 43, 6, 483-489 (2003)
[11] Jardin, S. C.; Breslau, J. A., Implicit solution of the four-field extended-magnetohydrodynamic equations using high-order high-continuity finite elements, Phys. Plasmas, 12, 5, 056101 (2005)
[12] Harned, D. S.; Kerner, W., Semi-implicit method for three-dimensional compressible magnetohydrodynamic simulation, J. Comput. Phys., 60, 62-75 (1985) · Zbl 0581.76057
[13] Schnack, D. D.; Barnes, D. C.; Harned, D. S.; Caramana, E. J., Semi-implicit magnetohydrodynamic calculations, J. Comput. Phys., 70, 330-354 (1987) · Zbl 0615.76109
[14] Harned, D. S.; Mikic, Z., Accurate semi-implicit treatment of the Hall effect in magnetohydrodynamic computations, J. Comput. Phys., 83, 1-15 (1989) · Zbl 0672.76051
[15] Sovinec, C. R.; Glasser, A. H.; Gianakon, T. A.; Barnes, D. C.; Nebel, R. A.; Kruger, S. E.; Schnack, D. D.; Plimpton, S. J.; Tarditi, A.; Chu, M. S., Nonlinear magnetohydrodynamics simulation using high-order finite elements, J. Comput. Phys., 195, 1, 355-386 (2004) · Zbl 1087.76070
[16] Hujeirat, A., IRMHD: an implicit radiative and magnetohydrodynamical solver for self-gravitating systems, Mon. Not. Roy. Astron. Soc., 298, 310-320 (1998)
[17] A.C. Robinson, et al., ALEGRA: An arbitrary Lagrangian-Eulerian multimaterial, multiphysics code, in: AIAA 2008-1235 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 2008.; A.C. Robinson, et al., ALEGRA: An arbitrary Lagrangian-Eulerian multimaterial, multiphysics code, in: AIAA 2008-1235 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 2008.
[18] G. Toth, R. Keppens, Versatile advection code, <http://www.phys.uu.nl/∼;toth/>; G. Toth, R. Keppens, Versatile advection code, <http://www.phys.uu.nl/∼;toth/>
[19] Knoll, D. A.; Chacón, L.; Margolin, L.; Mousseau, V. A., On balanced approximations for the time integration of multiple time-scale systems, J. Comput. Phys., 185, 583-611 (2003) · Zbl 1047.76074
[20] Hujeirat, A.; Rannacher, R., On the efficiency and robustness of implicit methods in computational astrophysics, New Astron. Rev., 45, 425-447 (2001)
[21] Reynolds, D. R.; Samtaney, R.; Woodward, C. S., A fully implicit numerical method for single-fluid resistive magnetohydrodynamics, J. Comput. Phys., 219, 1, 144-162 (2006) · Zbl 1103.76036
[22] Ovtchinnikov, S.; Dobrian, F.; Cai, X.-C.; Keyes, D., Additive Schwarz-based fully coupled implicit methods for resistive Hall magnetohydrodynamic problems, J. Comput. Phys., 225, 1919-1936 (2007) · Zbl 1343.76049
[23] Chacón, 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
[24] Chacón, L.; Knoll, D. A., A 2D high-\(β\) Hall MHD implicit nonlinear solver, J. Comput. Phys., 188, 2, 573-592 (2003) · Zbl 1127.76375
[25] Chacón, L., An optimal, parallel, fully implicit Newton-Krylov solver for three-dimensional visco-resistive magnetohydrodynamics, Phys. Plasmas, 15, 056103 (2008)
[26] Chacón, L., Scalable solvers for 3D magnetohydrodynamics, J. Phys: Conf. Ser., 125, 012041 (2008)
[27] Reynolds, D. R.; Samtaney, R.; Woodward, C. S., Operator-based preconditioning of stiff hyperbolic systems, SIAM J. Sci. Comput., 32, 150-170 (2010) · Zbl 1410.65090
[28] Salah, N. B.; Soulaimani, A.; Habashi, W. G.; Fortin, M., A conservative stabilized finite element method for the magnetohydrodynamics equations, Int. J. Num. Meth. Fuilds, 29, 535-554 (1999) · Zbl 0938.76049
[29] Brezzi, F., On existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO Model. Math. Anal. Numer., 21, 129-151 (1974) · Zbl 0338.90047
[30] Gunzburger, M., Finite Element Methods for Viscous Incompressible Flows (1989), Academic Press: Academic Press Boston · Zbl 0697.76031
[31] Barth, T.; Bochev, P.; Gunzburger, M.; Shadid, J., A taxonomy of consistently stabilized finite element methods for the Stokes problem, SIAM J. Sci. Comput., 25, 5, 1585-1607 (2004) · Zbl 1133.76307
[32] Brezzi, F.; Pitkaranta, J., On the stabilization of finite element approximations of the stokes problem, Efficient Solutions of Elliptic Systems, (Hackbusch, W., Vieweg Notes on Numerical Fluid Mechanics, 10 (1984), Vieweg: Vieweg Wiesbaden), 11-19 · Zbl 0552.76002
[33] Salah, N. B.; Soulaimani, A.; Habashi, W. G., A finite element method for magnetohydrodynamics, Comput. Meth. Appl. M., 190, 5867-5892 (2001) · Zbl 1044.76030
[34] Codina, R.; Hernandez-Silva, N., Stabilized finite element approximation of the stationary magneto-hydrodynamics equations, Comput. Mech., 38, 4-5, 344-355 (2006) · Zbl 1160.76025
[35] Gerbeau, J.-F., A stabilized finite element method for the incompressible magnetohydrodynamic equations, Numer. Math., 87, 83-111 (2000) · Zbl 0988.76050
[36] Lankalapalli, S.; Flaherty, J.; Shephard, M.; Strauss, H., An adaptive finite element method for magnetohydrodynamics, J. Comput. Phys., 225, 363-381 (2007) · Zbl 1118.76039
[37] Brown, P. N.; Saad, Y., Convergence theory of nonlinear Newton-Krylov algorithms, SIAM J. Optimiz., 4, 297-330 (1994) · Zbl 0814.65048
[38] Eisenstat, S.; Walker, H., Globally convergent inexact Newton methods, SIAM J. Optimiz., 4, 393-422 (1994) · Zbl 0814.65049
[39] Quarteroni, A.; Valli, A., Domain Decomposition Methods for Partial Differential Equations (1999), Oxford University Press: Oxford University Press Oxford · Zbl 0931.65118
[40] Sala, M.; Shadid, J. N.; Tuminaro, R. S., An improved convergence bound for aggregation-based domain decomposition preconditioners, SIAM J. Matrix Anal. A., 27, 3, 744-756 (2006) · Zbl 1105.65116
[41] Lin, P. T.; Sala, M.; Shadid, J. N.; Tuminaro, R. S., Performance of fully-coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport, Int. J. Numer. Meth. Eng., 67, 9, 208-225 (2006) · Zbl 1110.76315
[42] Strauss, H. R., Nonlinear, 3-dimensional magnetohydrodynamics of noncircular tokamaks, Phys. Fluids, 19, 1, 134-140 (1976)
[43] Hazeltine, R. D.; Kotschenreuther, M.; Morrison, P. J., A four-field model for tokamak plasma dynamics, Phys. Fluids, 28, 8, 2466-2477 (1985) · Zbl 0584.76124
[44] Drake, J. F.; Antonsen, T. M., Nonlinear reduced fluid equations for toroidal plasmas, Phys. Fluids, 27, 4, 898-908 (1984) · Zbl 0555.76097
[45] Moreau, R., Magnetohydrodynamics (1990), Kluwer: Kluwer Dordrecht · Zbl 0714.76003
[46] Davidson, P. A., An Introduction to Magnetohydrodynamics (2001), Cambridge University Press · Zbl 0974.76002
[47] Donea, J.; Huerta, A., Finite Element Methods for Flow Problems (2002), John Wiley
[48] Dedner, A.; Kemm, F.; Kroner, D.; Munz, C.-D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for the MHD equations, J. Comput. Phys., 175, 645-673 (2002) · Zbl 1059.76040
[49] Chacón, L., A non-staggered, conservative, ∇·B=0, finite-volume scheme for 3D implicit extended magnetohydrodynamics in curvilinear geometries, Comput. Phys. Commun., 163, 143-171 (2004) · Zbl 1196.76040
[50] Jackson, J. D., Classical Electrodynamics (1975), John Wiley & Sons · Zbl 0114.42903
[51] Brooks, A. N.; Hughes, T., Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Meth. Appl. M., 32, 199-259 (1982) · Zbl 0497.76041
[52] Hughes, T.; Franca, L.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Meth. Appl. M., 59, 85-99 (1986) · Zbl 0622.76077
[53] Hughes, T.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimentional advective-diffusive systems, Comput. Meth. Appl. M., 58, 305-328 (1986) · Zbl 0622.76075
[54] F. Shakib, Finite element analysis of the compressible Euler and Navier-Stokes equations, Ph.D. Thesis, Division of Applied Mathematics, Stanford University, 1989.; F. Shakib, Finite element analysis of the compressible Euler and Navier-Stokes equations, Ph.D. Thesis, Division of Applied Mathematics, Stanford University, 1989.
[55] Hughes, T., Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Meth. Appl. M., 127, 387-401 (1995) · Zbl 0866.76044
[56] Dennis, J. E.; Schnabel, R. B., Numerical methods for unconstrained optimization and nonlinear equations. Numerical methods for unconstrained optimization and nonlinear equations, Series in Automatic Computation (1983), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0579.65058
[57] Pawlowski, R. P.; Shadid, J. N.; Simonis, J. P.; Walker, H. F., Globalaization techniques for Newton-Krylov methods and applications to the fully-coupled solution of the Navier-Stokes equations, SIAM Rev., 48, 700-721 (2006) · Zbl 1110.65039
[58] Dembo, R. S.; Eisenstat, S. C.; Steihaug, T., Inexact Newton methods, SIAM J. Numer. Anal., 19, 400-408 (1982) · Zbl 0478.65030
[59] Eisenstat, S. C.; Walker, H. F., Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17, 16-32 (1996) · Zbl 0845.65021
[60] Heroux, M.; Bartlett, R.; Howle, V.; Hoekstra, R.; Hu, J.; Kolda, T.; Lehoucq, R.; Long, K.; Pawlowski, R.; Phipps, E.; Salinger, A.; Thornquist, H.; Tuminaro, R.; Willenbring, J.; Williams, A., An Overview of Trilinos Project, ACM Trans. Math. Software, 31, 3, 397-423 (2005) · Zbl 1136.65354
[61] Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM · Zbl 1002.65042
[62] Axelsson, O., Iterative Solution Methods (1994), Cambridge University Press: Cambridge University Press New York · Zbl 0795.65014
[63] Tuminaro, R.; Tong, C.; Shadid, J.; Devine, K. D.; Day, D., On a multilevel preconditioning module for unstructured mesh Krylov solvers: two-level Schwarz, Comm. Num. Method. Eng., 18, 383-389 (2002) · Zbl 0999.65101
[64] Shadid, J.; Tuminaro, R.; Devine, K.; Henningan, G.; Lin, P., Performance of fully-coupled domain decomposition preconditioners for finite element transport/reaction simulations, J. Comput. Phys., 205, 1, 24-47 (2005) · Zbl 1087.76069
[65] Trottenberg, U.; Oosterlee, C.; Schüller, A., Multigrid (2001), Academic Press: Academic Press London
[66] Shadid, J. N.; Salinger, A. G.; Pawlowski, R. P.; Lin, P. T.; Hennigan, G. L.; Tuminaro, R. S.; Lehoucq, R. B., Stabilized FE computational analysis of nonlinear steady state transport/reaction systems, Comput. Meth. Appl. M., 195, 1846-1871 (2006) · Zbl 1178.76240
[67] Ruge, J.; Stüben, K., Algebraic multigrid (AMG), (McCormick, S. F., Multigrid methods. Multigrid methods, Frontiers in Applied Mathematics, vol. 3 (1987), SIAM: SIAM Philadelphia, PA), 73-130
[68] G. Karypis, V. Kumar, Parallel multilevel \(k\); G. Karypis, V. Kumar, Parallel multilevel \(k\) · Zbl 0918.68073
[69] Vaněk, P.; Mandel, J.; Brezina, M., Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing, 56, 179-196 (1996) · Zbl 0851.65087
[70] Vaněk, P.; Brezina, M.; Mandel, J., Convergence of algebraic multigrid based on smoothed aggregation, Numer. Math., 88, 559-579 (2001) · Zbl 0992.65139
[71] Sala, M.; Tuminaro, R., A new Petrov-Galerkin smoothed aggregation preconditioner for nonsymmetric linear systems, SIAM J. Sci. Comput., 31, 1, 143-166 (2008) · Zbl 1183.76673
[72] M. Gee, C. Siefert, J. Hu, R. Tuminaro, M. Sala, ML 5.0 smoothed aggregation user’s guide, Technical Report. SAND2006-2649, Sandia National Laboratories, Albuquerque, NM 87185, 2006.; M. Gee, C. Siefert, J. Hu, R. Tuminaro, M. Sala, ML 5.0 smoothed aggregation user’s guide, Technical Report. SAND2006-2649, Sandia National Laboratories, Albuquerque, NM 87185, 2006.
[73] Davis, T., Direct methods for sparse linear systems (2006), SIAM: SIAM Philadelphia, PA · Zbl 1119.65021
[74] M. Sala, Amesos 2.0 reference guide, Technical Report. SAND2004-4820, Sandia National Laboratories, September 2004.; M. Sala, Amesos 2.0 reference guide, Technical Report. SAND2004-4820, Sandia National Laboratories, September 2004.
[75] Lin, P. T.; Sala, M.; Shadid, J. N.; Tuminaro, R. S., Performance of a geometric and an algebraic multilevel preconditioner for incompressible flow and transport, Computational Mechanics: WCCM VI in conjunction with APCOM’04 (2004), Tsinghus University Press/Springer-Verlag
[76] Lin, P. T.; Shadid, J. N.; Sala, M.; Tuminaro, R.; Hennigan, G. L.; Hoekstra, R. J., Performance of a parallel algebraic multilevel preconditioner for stabilized finite element semiconductor device modeling, J. Comput. Phys., 228, 17, 6079-6616 (2009)
[77] Moffat, H. K., Magnetic Field Generation in Electrically Conducting Fluids (1983), Cambridge University Press
[78] Codina, R.; Hernadez-Silva, N., Stabilized finite element approximation of the stationary magnetohydrodynamics equations, Comput. Mech., 38, 344-355 (2006) · Zbl 1160.76025
[79] Ainsworth, M.; Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis (2000), John Wiley · Zbl 1008.65076
[80] Batchelor, G. K., An Introduction to Fluid Mechanics (1967), Cambridge University Press · Zbl 0152.44402
[81] Fadeev, V. M.; Kvartskhava, I. F.; Komarov, N. N., Self-focusing of local plasma currents, Nucl. Fusion, 5, 3, 202-209 (1965)
[82] Hughes, M.; Pericleous, K. A.; Cross, M., The numerical modeling of dc electromagnetic pump and brake flow, Appl. Math. Model., 19, 713-723 (1995) · Zbl 0856.76093
[83] Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability (1961), Oxford University Press · Zbl 0142.44103
[84] Biskamp, D., Magnetic Reconnection in Plasmas (2000), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0891.76094
[85] Knoll, D. A.; Chacón, L., Coalescence of magnetic islands, sloshing, and the pressure problem, Phys. Plasmas, 13, 3, 32307-32311 (2006)
[86] Priest, E.; Forbes, T., Magnetic Reconnection: MHD Theory and Applications (2006), Cambridge University Press
[87] Shadid, J., A fully-coupled Newton-Krylov solution method for parallel unstructured finite element fluid flow, heat and mass transfer simulations, Int. J. CFD, 12, 199-211 (1999) · Zbl 0969.76049
[88] Bell, J. B.; Colella, P.; Glaz, H. M., A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85, 257 (1989) · Zbl 0681.76030
[89] Chorin, A. J., A numerical method for solving incompressible viscous problems, J. Comput. Phys., 2, 12 (1967) · Zbl 0149.44802
[90] Gresho, P. M., On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. part 1: Theory, Int. J. Num. Meth. Fluids, 11, 587-620 (1990) · Zbl 0712.76035
[91] Oran, E. S.; Boris, J. P., Numerical Simulation of Reactive Flow (2001), Cambridge Universty Press: Cambridge Universty Press Cambridge · Zbl 0762.76098
[92] Guermond, J. L.; Quartapelle, L., On stability of convergence of projection methods based on pressure poisson equation, Int. J. Numer. Meth. Fl., 26, 1039-1053 (1998) · Zbl 0912.76054
[93] Karniadakis, G. E.; Israeli, M.; Orszag, S. A., High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97, 414-443 (1991) · Zbl 0738.76050
[94] Perot, J. B., An analysis of the fractional step method, J. Comput. Phys., 108, 51-58 (1993) · Zbl 0778.76064
[95] Strikwerda, J. C.; Lee, Y. S., The accuracy of the fractional step method, SIAM J. Numer. Anal., 37, 1, 37-47 (1999) · Zbl 0953.65061
[96] Patankar, S. V., Numerical heat transfer and fluid flow (1980), Hemisphere Public Corporation: Hemisphere Public Corporation New York · Zbl 0595.76001
[97] Issa, R., Solution of the implicitly discretized fluid flows equations by operator splitting, J. Comp. Phys, 62, 1, 40-65 (1986) · Zbl 0619.76024
[98] Deng, G. B.; Piquet, J.; Queutey, P.; Visonneau, M. A., A new fully coupled solution of the Navier-Stokes equations, Int. J. Numer. Meth. Fl., 19, 605-639 (1994) · Zbl 0815.76054
[99] Deng, G. B.; Piquet, J.; Vasseur, X.; Visonneau, M. A., A new fully coupled method for computing turbulent flows, Comput. Fluids, 30, 445-472 (2001) · Zbl 1058.76042
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.