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

Lozinski, Alexei; Narski, Jacek; Negulescu, Claudia

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 6, pp. 1701-1724.

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.

Classification : 65N30, 65Z05, 35K20
Mots clés : anisotropic parabolic equation, ill-conditioned problem, singular perturbation model, limit model, asymptotic preserving scheme

[1] D. Aronson, The porous medium equation. Nonlinear Diffusion Problems, edited by A. Fasano, M. Primicerio. Lect. Notes Math. 1224 (1986) 1-46. | MR | Zbl

[2] S.F. Ashby, W.J. Bosl, R.D. Falgout, S.G. Smith, A.F. Tompson and T.J. Williams, 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] P. Basser and D. Jones, Diffusion-tensor mri: theory, experimental design and data analysis-a technical review. NMR Biomedicine 15 (2002) 456-467.

[4] C. Beaulieu, The basis of anisotropic water diffusion in the nervous system-a technical review. NMR Biomedicine 15 (2002) 435-455.

[5] B. Berkowitz, Characterizing flow and transport in fractured geological media: A review. Adv. Water Resources 25 (2002) 861-884.

[6] P. Degond, F. Deluzet, A. Lozinski, J. Narski and C. Negulescu, Duality-based asymptotic-preserving method for highly anisotropic diffusion equations. Commun. Math. Sci. 10 (2012) 1-31. | MR | Zbl

[7] P. Degond, A. Lozinski, J. Narski and C. Negulescu, An asymptotic-preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition. J. Comput. Phys. 231 (2012) 2724-2740. | MR

[8] Y. Dubinskii, Some integral inequalities and the solvability of degenerate quasi-linear elliptic systems of differential equations. Matematicheskii Sbornik 106 (1964) 458-480. | MR | Zbl

[9] Y. Dubinskii, Weak convergence for nonlinear elliptic and parabolic equations. Matematicheskii Sbornik 109 (1965) 609-642. | MR | Zbl

[10] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Stud. Adv. Math. CRC press (1992). | MR | Zbl

[11] S. Günter, K. Lackner C. Tichmann, Finite element and higher order difference formulations for modelling heat transport in magnetised plasmas. J. Comput. Phys. 226 (2007) 2306-2316.

[12] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Ser. Comput. Math. Springer-Verlag, New York (1987). | Zbl

[13] H. Jian and B. Song, Solutions of the anisotropic porous medium equation in Rn under an l1-initial value. Nonlinear Anal. 64 (2006) 2098-2111. | MR | Zbl

[14] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441-454. | MR | Zbl

[15] J. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villars (1969). | Zbl

[16] J.-L. Lions and E. Magenes, 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 | Zbl

[17] H. Lutjens and J. Luciani, The xtor code for nonlinear 3d simulations of mhd instabilities in tokamak plasmas. J. Comput. Phys. 227 (2008) 6944-6966. | MR

[18] A. Mentrelli and C. Negulescu, Asymptotic preserving scheme for highly anisotropic, nonlinear diffusion equations J. Comput. Phys. 231 (2012) 8229-8245. | MR

[19] W. Park, E. Belova, G. Fu, X. Tang, H. Strauss L. Sugiyama, Plasma simulation studies using multilevel physics models. Phys. Plasmas 6 (1999) 1796.

[20] P. Perona and J. Malik, 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] J. Narski, Anisotropic finite elements with high aspect ratio for an Asymptotic Preserving method for highly anisotropic elliptic equation. Preprint arXiv:1302.4269 (2013).

[23] J. Narski and M. Ottaviani, Asymptotic Preserving scheme for strongly anisotropic parabolic equations for arbitrary anisotropy direction. Preprint arXiv:1303.5219 (2013).

[24] J. Simon, Compact sets in the space Lp(0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65-96. | MR | Zbl

[25] P. Tamain, Etude des flux de matière dans le plasma de bord des tokamaks. Ph.D. Thesis, Marseille 1 (2007).

[26] J. Vázquez, The porous medium equation: mathematical theory. Oxford University Press, USA (2007). | Zbl

[27] J. Weickert, Anisotropic diffusion in image processing. European Consortium for Mathematics in Industry. B.G. Teubner, Stuttgart (1998). | MR | Zbl

[28] J. Wesson, Tokamaks. Oxford University Press, New York (1987). | Zbl

[29] O.C. Zienkiewicz and R.L. Taylor, The finite element method. Vol. 1. Butterworth-Heinemann, Oxford (2000). | MR | Zbl

[30] J. Wloka, Partial diflerential equations. Cambridge University Press (1987). | MR | Zbl