A Hybrid Method for Anisotropic Elliptic Problems Based on the Coupling of an Asymptotic-Preserving Method with the Asymptotic Limit Model

This paper presents a hybrid numerical method to solve efficiently a class of highly anisotropic elliptic problems. The anisotropy is aligned with one coordinate-axis and its strength is described by a parameter $\eps \in (0,1]$, which can largely vary in the study domain. Our hybrid model is based on asymptotic techniques and couples (spatially) an Asymptotic-Preserving model with its asymptotic Limit model, the latter being used in regions where the anisotropy parameter $\eps$ is small. Adequate coupling conditions link the two models. Aim of this hybrid procedure is to reduce the computational time for problems where the region of small $\eps$-values extends over a significant part of the domain, and this due to the reduced complexity of the limit model.

[1]  Joseph P. Dougherty,et al.  Waves in plasmas. , 1993 .

[2]  Walter J. Heikkila,et al.  Earth's Magnetosphere: Formed by the Low-Latitude Boundary Layer , 2020 .

[3]  Christophe Besse,et al.  Efficient Numerical Methods for Strongly Anisotropic Elliptic Equations , 2013, J. Sci. Comput..

[4]  Stephen C. Jardin,et al.  Review of implicit methods for the magnetohydrodynamic description of magnetically confined plasmas , 2012, J. Comput. Phys..

[5]  J. Huba,et al.  Three-Dimensional Modeling of Equatorial Spread F , 2011 .

[6]  Francis F. Chen,et al.  Introduction to Plasma Physics and Controlled Fusion , 2015 .

[7]  Chang-Ming Yang,et al.  Analyse et mise en oeuvre des schémas numériques pour la physique des plasmas ionosphériques et de tokamaks , 2011 .

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

[10]  P. Gibbon,et al.  Introduction to Plasma Physics , 2017, 2007.04783.

[11]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[12]  Luc Mieussens,et al.  A smooth transition model between kinetic and hydrodynamic equations , 2005 .

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

[14]  S. Zalesak,et al.  Three-dimensional simulation of equatorial spread-F with meridional wind effects , 2009 .

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

[16]  Pierre Degond,et al.  Kinetic boundary layers and fluid-kinetic coupling in semiconductors , 1999 .

[17]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

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

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

[20]  Axel Klar,et al.  Transition from Kinetic theory to macroscopic fluid equations: A problem for domain decomposition and a source for new algorithms , 2000 .

[21]  Axel Klar,et al.  Asymptotic-Induced Domain Decomposition Methods for Kinetic and Drift Diffusion Semiconductor Equations , 1998, SIAM J. Sci. Comput..

[22]  George Karypis,et al.  A Software Package for Partitioning Unstructured Graphs , Partitioning Meshes , and Computing Fill-Reducing Orderings of Sparse Matrices Version 5 . 0 , 1998 .

[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]  Shi Jin,et al.  A Smooth Transition Model between Kinetic and Diffusion Equations , 2004, SIAM J. Numer. Anal..

[25]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[26]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[27]  Fabrice Deluzet,et al.  Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations , 2010, 1008.3405.

[28]  Robert W. Schunk,et al.  Ionospheres : physics, plasma physics, and chemistry , 2000 .

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

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