This paper deals with the numerical study of a nonlinear, strongly anisotropic heat equation. The use of standard schemes in this situation leads to poor results, due to the high anisotropy. An Asymptotic-Preserving method is introduced in this paper, which is second-order accurate in both, temporal and spacial variables. The discretization in time is done using an L-stable Runge-Kutta scheme. The convergence of the method is shown to be independent of the anisotropy parameter , and this for fixed coarse Cartesian grids and for variable anisotropy directions. The context of this work are magnetically confined fusion plasmas.
Mots-clés : anisotropic parabolic equation, ill-conditioned problem, singular perturbation model, limit model, asymptotic preserving scheme
@article{M2AN_2014__48_6_1701_0, author = {Lozinski, Alexei and Narski, Jacek and Negulescu, Claudia}, title = {Highly anisotropic nonlinear temperature balance equation and its numerical solution using asymptotic-preserving schemes of second order in time}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1701--1724}, publisher = {EDP-Sciences}, volume = {48}, number = {6}, year = {2014}, doi = {10.1051/m2an/2014016}, mrnumber = {3264370}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an/2014016/} }
TY - JOUR AU - Lozinski, Alexei AU - Narski, Jacek AU - Negulescu, Claudia TI - Highly anisotropic nonlinear temperature balance equation and its numerical solution using asymptotic-preserving schemes of second order in time JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 1701 EP - 1724 VL - 48 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2014016/ DO - 10.1051/m2an/2014016 LA - en ID - M2AN_2014__48_6_1701_0 ER -
%0 Journal Article %A Lozinski, Alexei %A Narski, Jacek %A Negulescu, Claudia %T Highly anisotropic nonlinear temperature balance equation and its numerical solution using asymptotic-preserving schemes of second order in time %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 1701-1724 %V 48 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2014016/ %R 10.1051/m2an/2014016 %G en %F M2AN_2014__48_6_1701_0
Lozinski, Alexei; Narski, Jacek; Negulescu, Claudia. Highly anisotropic nonlinear temperature balance equation and its numerical solution using asymptotic-preserving schemes of second order in time. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 6, pp. 1701-1724. doi : 10.1051/m2an/2014016. http://archive.numdam.org/articles/10.1051/m2an/2014016/
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