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.
@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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {48},
year = {2014},
pages = {1701-1724},
doi = {10.1051/m2an/2014016},
mrnumber = {3264370},
language = {en},
url = {http://dml.mathdoc.fr/item/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 - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) pp. 1701-1724. doi : 10.1051/m2an/2014016. http://gdmltest.u-ga.fr/item/M2AN_2014__48_6_1701_0/
[1] , The porous medium equation. Nonlinear Diffusion Problems, edited by A. Fasano, M. Primicerio. Lect. Notes Math. 1224 (1986) 1-46. | MR 877986 | Zbl 0626.76097
[2] , , , , and , A Numerical Simulation of Groundwater Flow and Contaminant Transport on the CRAY T3D and C90 Supercomputers. Int. J. High Performance Comput. Appl. 13 (1999) 80-93.
[3] and , Diffusion-tensor mri: theory, experimental design and data analysis-a technical review. NMR Biomedicine 15 (2002) 456-467.
[4] , The basis of anisotropic water diffusion in the nervous system-a technical review. NMR Biomedicine 15 (2002) 435-455.
[5] , Characterizing flow and transport in fractured geological media: A review. Adv. Water Resources 25 (2002) 861-884.
[6] , , , and , Duality-based asymptotic-preserving method for highly anisotropic diffusion equations. Commun. Math. Sci. 10 (2012) 1-31. | MR 2901299 | Zbl 1272.65090
[7] , , and , An asymptotic-preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition. J. Comput. Phys. 231 (2012) 2724-2740. | MR 2882095 | Zbl pre06036625
[8] , Some integral inequalities and the solvability of degenerate quasi-linear elliptic systems of differential equations. Matematicheskii Sbornik 106 (1964) 458-480. | MR 168904 | Zbl 0174.42501
[9] , Weak convergence for nonlinear elliptic and parabolic equations. Matematicheskii Sbornik 109 (1965) 609-642. | MR 190546 | Zbl 0177.37403
[10] and , Measure theory and fine properties of functions. Stud. Adv. Math. CRC press (1992). | MR 1158660 | Zbl 0804.28001
[11] , C. Tichmann, Finite element and higher order difference formulations for modelling heat transport in magnetised plasmas. J. Comput. Phys. 226 (2007) 2306-2316. | Zbl pre05207665
[12] and , Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Ser. Comput. Math. Springer-Verlag, New York (1987). | Zbl 0729.65051
[13] and , Solutions of the anisotropic porous medium equation in Rn under an l1-initial value. Nonlinear Anal. 64 (2006) 2098-2111. | MR 2211202 | Zbl 1100.35059
[14] , Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441-454. | MR 1718639 | Zbl 0947.82008
[15] , Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villars (1969). | Zbl 0189.40603
[16] and , Non-homogeneous boundary value problems and applications. Vol. I. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York (1972). | MR 350177 | Zbl 0223.35039
[17] and , The xtor code for nonlinear 3d simulations of mhd instabilities in tokamak plasmas. J. Comput. Phys. 227 (2008) 6944-6966. | MR 2435437 | Zbl pre05303093
[18] and , Asymptotic preserving scheme for highly anisotropic, nonlinear diffusion equations J. Comput. Phys. 231 (2012) 8229-8245. | MR 2979850
[19] , , , , L. Sugiyama, Plasma simulation studies using multilevel physics models. Phys. Plasmas 6 (1999) 1796.
[20] and , Scale-space and edge detection using anisotropic diffusion. Pattern Analysis and Machine Intelligence, IEEE Trans. 12 (1990) 629-639.
[21] M. Pierre, personal e-mail (2011).
[22] , Anisotropic finite elements with high aspect ratio for an Asymptotic Preserving method for highly anisotropic elliptic equation. Preprint arXiv:1302.4269 (2013).
[23] and , Asymptotic Preserving scheme for strongly anisotropic parabolic equations for arbitrary anisotropy direction. Preprint arXiv:1303.5219 (2013).
[24] , Compact sets in the space Lp(0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65-96. | MR 916688 | Zbl 0629.46031
[25] , Etude des flux de matière dans le plasma de bord des tokamaks. Ph.D. Thesis, Marseille 1 (2007).
[26] , The porous medium equation: mathematical theory. Oxford University Press, USA (2007). | Zbl 1107.35003
[27] , Anisotropic diffusion in image processing. European Consortium for Mathematics in Industry. B.G. Teubner, Stuttgart (1998). | MR 1666943 | Zbl 0886.68131
[28] , Tokamaks. Oxford University Press, New York (1987). | Zbl 1111.82054
[29] and , The finite element method. Vol. 1. Butterworth-Heinemann, Oxford (2000). | MR 1897985 | Zbl 0991.74002
[30] , Partial diflerential equations. Cambridge University Press (1987). | MR 895589 | Zbl 0623.35006