Highly anisotropic nonlinear temperature balance equation and its numerical solution using asymptotic-preserving schemes of second order in time ∗
1 Université de Toulouse, UPS, INSA,
UT1, UTM, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062
2 Laboratoire de Mathematiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon cedex, France.
Revised: 14 February 2014
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.
Mathematics Subject Classification: 65N30 / 65Z05 / 35K20
Key words: Anisotropic parabolic equation / Ill-conditioned problem / singular perturbation model / limit model / asymptotic preserving scheme
© EDP Sciences, SMAI, 2014