Asymptotic-preserving scheme for highly anisotropic non-linear diffusion equations

Heat transfer in magnetically confined plasmas is a process characterized by non-linear and extremely high anisotropic diffusion phenomena. Standard numerical methods, successfully employed in the numerical treatment of classical diffusion problems, are generally inefficient, or even prone to break down, when such high anisotropies come into play, leading thus to the need of new numerical techniques suitable for this kind of problems. In the present paper, the authors propose a numerical scheme based on an asymptotic-preserving (AP) reformulation of this non-linear evolution problem, generalizing the ideas introduced in a previous paper for the case of elliptic anisotropic problems [P. Degond, A. Lozinski, J. Narski, C. Negulescu, An asymptotic-preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition, J. Comput. Phys. 231 (7) (2012) 2724-2740]. The performances of the here proposed AP scheme are tested numerically; in particular it is shown that the scheme is capable to deal with problems characterized by a high degree of anisotropy, thus proving to be suitable for the study of anisotropic diffusion in magnetically confined plasmas.

[1]  Claes Johnson.  Numerical solution of partial differential equations by the finite element method , 1988 .

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

[3]  P. Degond.  Asymptotic-Preserving Schemes for Fluid Models of Plasmas , 2011, 1104.1869.

[4]  Prateek Sharma,et al.  A fast semi-implicit method for anisotropic diffusion , 2010, J. Comput. Phys..

[5]  Timothy A. Davis,et al.  A combined unifrontal/multifrontal method for unsymmetric sparse matrices , 1999, TOMS.

[6]  Derek K. Jones,et al.  Diffusion‐tensor MRI: theory, experimental design and data analysis – a technical review , 2002 .

[7]  B. Berkowitz.  Characterizing flow and transport in fractured geological media: A review , 2002 .

[8]  Fabrice Deluzet,et al.  Numerical approximation of the Euler-Maxwell model in the quasineutral limit , 2011, J. Comput. Phys..

[9]  J. Vázquez.  The Porous Medium Equation: Mathematical Theory , 2006 .

[10]  N. Crouseilles,et al.  An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits. , 2011 .

[11]  Sergey Shmarev,et al.  A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions , 2005 .

[12]  Isabelle Ramière,et al.  Convergence analysis of the Q1‐finite element method for elliptic problems with non‐boundary‐fitted meshes , 2008 .

[13]  Jean-Luc Guermond,et al.  Eléments finis : théorie, applications, mise en œuvre , 2002 .

[14]  C. Beaulieu,et al.  The basis of anisotropic water diffusion in the nervous system – a technical review , 2002, NMR in biomedicine.

[15]  Claudia Negulescu,et al.  Numerical study of a nonlinear heat equation for plasma physics , 2011, Int. J. Comput. Math..

[16]  Livia Isoardi.  Modelisation du transport dans le plasma de bord d'un tokamak , 2010 .

[17]  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..

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

[19]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

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

[21]  Timothy A. Davis,et al.  A column pre-ordering strategy for the unsymmetric-pattern multifrontal method , 2004, TOMS.

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

[23]  Claudia Negulescu,et al.  An asymptotic-preserving method for highly anisotropic elliptic equations based on a Micro-Macro decomposition , 2011, J. Comput. Phys..

[24]  Luc Mieussens,et al.  Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics , 2008, J. Comput. Phys..

[25]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[26]  Timothy A. Davis,et al.  An Unsymmetric-pattern Multifrontal Method for Sparse Lu Factorization , 1993 .

[27]  Luc Mieussens,et al.  A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit , 2008, SIAM J. Sci. Comput..

[28]  J. Vázquez.  The Porous Medium Equation , 2006 .

[29]  Philippe Ghendrih,et al.  The Plasma Boundary of Magnetic Fusion Devices , 2001 .

[30]  Fabrice Deluzet,et al.  Asymptotic-preserving scheme for a bi-fluid Euler-Lorentz model , 2011, 1104.3339.