Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations

The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preserving method generalizes the method investigated in a previous paper [arXiv:0903.4984v2] to the case of an arbitrary anisotropy direction field.

[1]  Thierry Gallouët,et al.  A new finite volume scheme for anisotropic diffusion problems on general grids: convergence analysis , 2007 .

[2]  Sidney L. Ossakow,et al.  Three‐dimensional nonlinear evolution of equatorial ionospheric spread‐F bubbles , 2003 .

[3]  Jens Markus Melenk,et al.  hp-Finite Element Methods for Singular Perturbations , 2002 .

[4]  Kenro Miyamoto,et al.  Controlled Fusion and Plasma Physics , 2006 .

[5]  T. Manku,et al.  Electrical properties of silicon under nonuniform stress , 1993 .

[6]  Timothy J. Williams,et al.  A Numerical Simulation of Groundwater Flow and Contaminant Transport on the CRAY T3D and C90 Supercomputers , 1999, Int. J. High Perform. Comput. Appl..

[7]  Dimits.  Fluid simulations of tokamak turbulence in quasiballooning coordinates. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  Shi Jin,et al.  Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations , 1999, SIAM J. Sci. Comput..

[9]  M. Ottaviani,et al.  An alternative approach to field-aligned coordinates for plasma turbulence simulations , 2010, 1002.0748.

[10]  P. Degond,et al.  A MODEL HIERARCHY FOR IONOSPHERIC PLASMA MODELING , 2004 .

[11]  L. S. Pontryagin,et al.  Ordinary Differential Equations , 1963 .

[12]  Prateek Sharma,et al.  Preserving monotonicity in anisotropic diffusion , 2007, J. Comput. Phys..

[13]  Fabrice Deluzet,et al.  An Asymptotic Preserving Scheme for Strongly Anisotropic Elliptic Problems , 2009, Multiscale Model. Simul..

[14]  S. Hamada,et al.  Hydromagnetic equilibria and their proper coordinates , 1962 .

[15]  David P. Stern,et al.  Geomagnetic Euler potentials , 1967 .

[16]  Michael C. Kelley,et al.  Mid-latitude ionospheric fluctuation spectra due to secondary E×B instabilities , 2004 .

[17]  Xiangchu Feng,et al.  Anisotropic Diffusion with Nonlinear Structure Tensor , 2008, Multiscale Model. Simul..

[18]  X. Garbet,et al.  Computing ITG turbulence with a full-f semi-Lagrangian code , 2008 .

[19]  Gabriel Wittum,et al.  Robust Schur complement method for strongly anisotropic elliptic equations , 1999, Numer. Linear Algebra Appl..

[20]  Jonathan J. Hu,et al.  A new smoothed aggregation multigrid method for anisotropic problems , 2007, Numer. Linear Algebra Appl..

[21]  Ignacio M. Llorente,et al.  Robust multigrid smoothers for three dimensional elliptic equations with strong anisotropies , 1998 .

[22]  A. Boozer.  Establishment of magnetic coordinates for a given magnetic field , 1982 .

[23]  Gabriel Wittum,et al.  Robust Schur complement method for strongly anisotropic elliptic equations , 1999 .

[24]  Christophe Le Potier,et al.  Schéma volumes finis pour des opérateurs de diffusion fortement anisotropes sur des maillages non structurés , 2005 .

[25]  W. D. D'haeseleer,et al.  Flux coordinates and magnetic field structure : a guide to a fundamental tool of plasma theory , 1991 .

[26]  Fabrice Deluzet,et al.  Asymptotic-Preserving Particle-In-Cell method for the Vlasov-Poisson system near quasineutrality , 2010, J. Comput. Phys..

[27]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[28]  Gregory W. Hammett,et al.  Field‐aligned coordinates for nonlinear simulations of tokamak turbulence , 1995 .

[29]  Patrick Amestoy,et al.  A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling , 2001, SIAM J. Matrix Anal. Appl..

[30]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[31]  M. Shashkov,et al.  The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes , 2006 .

[32]  Y. Igitkhanov,et al.  Fluid Description of Edge Plasma Transport in a Non-Orthogonal Coordinate System , 1994 .

[33]  Pierre Degond,et al.  An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit , 2007, J. Comput. Phys..

[34]  L. Garrigues,et al.  Physics, simulation and diagnostics of Hall effect thrusters , 2008 .

[35]  V. Rozhansky,et al.  Poloidal and toroidal flows in tokamak plasma near magnetic islands , 2004 .

[36]  Luis Chacon,et al.  An optimal, parallel, fully implicit Newton–Krylov solver for three-dimensional viscoresistive magnetohydrodynamicsa) , 2008 .

[37]  Y. Notay.  An aggregation-based algebraic multigrid method , 2010 .

[38]  L. Giraud,et al.  Schur complement preconditioners for anisotropic problems , 1999 .

[39]  M. J. Keskinen,et al.  Nonlinear theory of the E × B instability with an inhomogeneous electric field , 1984 .

[40]  Daniil Svyatskiy,et al.  A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems , 2009, J. Comput. Phys..

[41]  Fabrice Deluzet,et al.  An Asymptotic Preserving scheme for the Euler equations in a strong magnetic field , 2009, J. Comput. Phys..

[42]  Sibylle Günter,et al.  Modelling of heat transport in magnetised plasmas using non-aligned coordinates , 2005 .