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Highly anisotropic nonlinear temperature balance equation and its numerical solution using asymptotic-preserving schemes of second order in time ∗

Published online by Cambridge University Press:  26 September 2014

Alexei Lozinski ,
Jacek Narski  and
Claudia Negulescu
Alexei Lozinski
Affiliation:
Université de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France.. jacek.narski@math.univ-toulouse.fr; claudia.negulescu@math.univ-toulouse.fr Laboratoire de Mathematiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon cedex, France.; alexei.lozinski@univ-fcomte.fr
Jacek Narski
Affiliation:
Université de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France.. jacek.narski@math.univ-toulouse.fr; claudia.negulescu@math.univ-toulouse.fr
Claudia Negulescu
Affiliation:
Université de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France.. jacek.narski@math.univ-toulouse.fr; claudia.negulescu@math.univ-toulouse.fr
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Abstract

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.

Keywords

Anisotropic parabolic equationIll-conditioned problemsingular perturbation modellimit modelasymptotic preserving scheme
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 6 , November 2014 , pp. 1701 - 1724
Copyright
© EDP Sciences, SMAI, 2014

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