Numerical simulation of four-field extended magnetohydrodynamics in dynamically adaptive curvilinear coordinates via Newton-Krylov-Schwarz

Numerical simulations of the four-field extended magnetohydrodynamics (MHD) equations with hyper-resistivity terms present a difficult challenge because of demanding spatial resolution requirements. A time-dependent sequence of r-refinement adaptive grids obtained from solving a single Monge-Ampere (MA) equation addresses the high-resolution requirements near the x-point for numerical simulation of the magnetic reconnection problem. The MHD equations are transformed from Cartesian coordinates to solution-defined curvilinear coordinates. After the application of an implicit scheme to the time-dependent problem, the parallel Newton-Krylov-Schwarz (NKS) algorithm is used to solve the system at each time step. Convergence and accuracy studies show that the curvilinear solution requires less computational effort than a pure Cartesian treatment. This is due both to the more optimal placement of the grid points and to the improved convergence of the implicit solver, nonlinearly and linearly. The latter effect, which is significant (more than an order of magnitude in number of inner linear iterations for equivalent accuracy), does not yet seem to be widely appreciated.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  A. H. Glasser,et al.  Specification of monitor metrics for generating vector field-aligned numerical grids , 2005 .

[3]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[4]  Francesco Porcelli,et al.  Hamiltonian magnetic reconnection , 1999 .

[5]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[6]  David E. Keyes,et al.  Moving grids for magnetic reconnection via Newton-Krylov methods , 2011, Comput. Phys. Commun..

[7]  Xuefei Yuan.  Grid Generation Techniques in Magnetic Reconnection Simulation , 2011 .

[8]  Luis A. Caffarelli,et al.  The Dirichlet problem for nonlinear second-order elliptic equations I , 1984 .

[9]  J. Brackbill.  An adaptive grid with directional control , 1993 .

[10]  David E. Keyes,et al.  Application of PDSLin to the magnetic reconnection problem , 2013 .

[11]  S. C. Jardin,et al.  Global Extended MHD Studies of Fast Magnetic Reconnection , 2002 .

[12]  C. D. Boor,et al.  Good approximation by splines with variable knots. II , 1974 .

[13]  Joe F. Thompson,et al.  Numerical grid generation: Foundations and applications , 1985 .

[14]  L. Giraud,et al.  A note on relaxed and flexible GMRES , 2004 .

[15]  Stephen C. Jardin,et al.  A high-order implicit finite element method for integrating the two-fluid magnetohydrodynamic equations in two dimensions , 2007, J. Comput. Phys..

[16]  Gian Luca Delzanno,et al.  Robust, multidimensional mesh-motion based on Monge-Kantorovich equidistribution , 2011, J. Comput. Phys..

[17]  Xiao-Chuan Cai,et al.  A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems , 1999, SIAM J. Sci. Comput..

[18]  J. Scanlon.  The Dirichlet problem for nonlinear elliptic equations. , 1955 .

[19]  Gian Luca Delzanno,et al.  An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization , 2008, J. Comput. Phys..

[20]  Kazunari Shibata,et al.  New observational facts about solar flares from Yohkoh studies — Evidence of magnetic reconnection and a unified model of flares , 1996 .

[21]  J. Breslau,et al.  Global extended magnetohydrodynamic studies of fast magnetic reconnection , 2003 .

[22]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[23]  Vladimir D. Liseikin,et al.  A Computational Differential Geometry Approach to Grid Generation (Scientific Computation) , 2003 .

[24]  William Gropp,et al.  Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries , 1997, SciTools.

[25]  Dana A. Knoll,et al.  An Implicit, Nonlinear Reduced Resistive MHD Solver , 2002 .

[26]  Jun Zou,et al.  Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes , 1994, Numerical Algorithms.

[27]  Daniel N. Baker,et al.  Neutral line model of substorms: Past results and present view , 1996 .

[28]  R. Fitzpatrick,et al.  Scaling of forced magnetic reconnection in the Hall-magnetohydrodynamical Taylor problem with arbitrary guide field , 2004 .

[29]  A. A. Amsden,et al.  A numerical fluid dynamics calculation method for all flow speeds , 1971 .

[30]  Michael Hesse,et al.  Geospace Environmental Modeling (GEM) magnetic reconnection challenge , 2001 .

[31]  Dieter Biskamp,et al.  Magnetic reconnection in plasmas , 2000 .

[32]  Alfio Quarteroni,et al.  Domain Decomposition Methods for Partial Differential Equations , 1999 .

[33]  Homer F. Walker,et al.  Choosing the Forcing Terms in an Inexact Newton Method , 1996, SIAM J. Sci. Comput..

[34]  Shrinivas Lankalapalli,et al.  An adaptive finite element method for magnetohydrodynamics , 1998, J. Comput. Phys..

[35]  William H. Press,et al.  Numerical recipes in C , 2002 .

[36]  Joseph E. Flaherty,et al.  An adaptive mesh-moving and local refinement method for time-dependent partial differential equations , 1990, TOMS.

[37]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[38]  David E. Keyes,et al.  Nonlinearly Preconditioned Inexact Newton Algorithms , 2002, SIAM J. Sci. Comput..

[39]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[40]  Barry F. Smith,et al.  Newton-Krylov-Schwarz methods for aerodynamics problems : compressible and incompressible flows on unstructured grids. , 1999 .

[41]  M. Shashkov.  Conservative Finite-Difference Methods on General Grids , 1996 .

[42]  A. H. Glasser,et al.  Some computational aspects on generating numerical grids , 2006 .

[43]  R. Chandra.  Conjugate gradient methods for partial differential equations. , 1978 .

[44]  James Drake,et al.  Two-fluid theory of collisionless magnetic reconnection , 1997 .

[45]  David M. Young,et al.  Applied Iterative Methods , 2004 .

[46]  R. Fitzpatrick,et al.  Scaling of forced magnetic reconnection in the Hall-magnetohydrodynamic Taylor problem , 2004 .

[47]  Dana A. Knoll,et al.  Enhanced Nonlinear Iterative Techniques Applied to a Nonequilibrium Plasma Flow , 1998, SIAM J. Sci. Comput..

[48]  David E. Keyes,et al.  Additive Schwarz-based fully coupled implicit methods for resistive Hall magnetohydrodynamic problems , 2007, J. Comput. Phys..

[49]  Stephen C. Jardin,et al.  Implicit solution of the four-field extended-magnetohydrodynamic equations using high-order high-continuity finite elementsa) , 2004 .

[50]  A. A. Amsden,et al.  Numerical calculation of almost incompressible flow , 1968 .

[51]  L. Kantorovich.  On the Translocation of Masses , 2006 .

[52]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[53]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[54]  Vladimir D. Liseikin,et al.  Grid Generation Methods , 1999 .

[55]  Stephen C. Jardin,et al.  Computational Methods in Plasma Physics , 2010 .

[56]  M. D. Tidriri,et al.  Development and Study of Newton-Krylov-Schwarz Algorithms , 2001 .