Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T19:19:34.498Z Has data issue: false hasContentIssue false
  • English
  • Français

Highly anisotropic nonlinear temperature balance equation andits 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.. [email protected]; [email protected] Laboratoire de Mathematiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon cedex, France.; [email protected]
Jacek Narski
Affiliation:
Université de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France.. [email protected]; [email protected]
Claudia Negulescu
Affiliation:
Université de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France.. [email protected]; [email protected]
Get access

Abstract

This paper deals with the numerical study of a nonlinear, strongly anisotropic heatequation. The use of standard schemes in this situation leads to poor results, due to thehigh anisotropy. An Asymptotic-Preserving method is introduced in this paper, which issecond-order accurate in both, temporal and spacial variables. The discretization in timeis done using an L-stable Runge−Kutta scheme. The convergence of the method is shown to beindependent of the anisotropy parameter , andthis for fixed coarse Cartesian grids and for variable anisotropy directions. The contextof 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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

D. Aronson, The porous medium equation. Nonlinear Diffusion Problems, edited by A. Fasano, M. Primicerio. Lect. Notes Math. 1224 (1986) 1–46.
Ashby, S.F., Bosl, W.J., Falgout, R.D., Smith, S.G., Tompson, A.F. and Williams, T.J., 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. Google Scholar
Basser, P. and Jones, D., Diffusion-tensor mri: theory, experimental design and data analysis–a technical review. NMR Biomedicine 15 (2002) 456–467. Google ScholarPubMed
Beaulieu, C., The basis of anisotropic water diffusion in the nervous system–a technical review. NMR Biomedicine 15 (2002) 435–455. Google ScholarPubMed
Berkowitz, B., Characterizing flow and transport in fractured geological media: A review. Adv. Water Resources 25 (2002) 861–884. Google Scholar
Degond, P., Deluzet, F., Lozinski, A., Narski, J. and Negulescu, C., Duality-based asymptotic-preserving method for highly anisotropic diffusion equations. Commun. Math. Sci. 10 (2012) 1–31. Google Scholar
Degond, P., Lozinski, A., Narski, J. and Negulescu, C., An asymptotic-preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition. J. Comput. Phys. 231 (2012) 2724–2740. Google Scholar
Dubinskii, Y., Some integral inequalities and the solvability of degenerate quasi-linear elliptic systems of differential equations. Matematicheskii Sbornik 106 (1964) 458–480. Google Scholar
Dubinskii, Y., Weak convergence for nonlinear elliptic and parabolic equations. Matematicheskii Sbornik 109 (1965) 609–642. Google Scholar
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Stud. Adv. Math. CRC press (1992).
Günter, S., Lackner, K. Tichmann, C., Finite element and higher order difference formulations for modelling heat transport in magnetised plasmas. J. Comput. Phys. 226 (2007) 2306–2316. Google Scholar
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Ser. Comput. Math. Springer-Verlag, New York (1987).
Jian, H. and Song, B., Solutions of the anisotropic porous medium equation in Rn under an l 1-initial value. Nonlinear Anal. 64 (2006) 2098–2111. Google Scholar
Jin, S., Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441–454. Google Scholar
J. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villars (1969).
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).
Lutjens, H. and Luciani, J., The xtor code for nonlinear 3d simulations of mhd instabilities in tokamak plasmas. J. Comput. Phys. 227 (2008) 6944–6966. Google Scholar
Mentrelli, A. and Negulescu, C., Asymptotic preserving scheme for highly anisotropic, nonlinear diffusion equations J. Comput. Phys. 231 (2012) 8229–8245. Google Scholar
Park, W., Belova, E., Fu, G., Tang, X., Strauss, H. Sugiyama, L., Plasma simulation studies using multilevel physics models. Phys. Plasmas 6 (1999) 1796. Google Scholar
Perona, P. and Malik, J., Scale-space and edge detection using anisotropic diffusion. Pattern Analysis and Machine Intelligence, IEEE Trans. 12 (1990) 629–639. CrossRefGoogle Scholar
M. Pierre, personal e-mail (2011).
J. Narski, Anisotropic finite elements with high aspect ratio for an Asymptotic Preserving method for highly anisotropic elliptic equation. Preprint arXiv:1302.4269 (2013).
J. Narski and M. Ottaviani, Asymptotic Preserving scheme for strongly anisotropic parabolic equations for arbitrary anisotropy direction. Preprint arXiv:1303.5219 (2013).
Simon, J., Compact sets in the space L p(0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65–96. Google Scholar
P. Tamain, Etude des flux de matière dans le plasma de bord des tokamaks. Ph.D. Thesis, Marseille 1 (2007).
J. Vázquez, The porous medium equation: mathematical theory. Oxford University Press, USA (2007).
J. Weickert, Anisotropic diffusion in image processing. European Consortium for Mathematics in Industry. B.G. Teubner, Stuttgart (1998).
J. Wesson, Tokamaks. Oxford University Press, New York (1987).
O.C. Zienkiewicz and R.L. Taylor, The finite element method. Vol. 1. Butterworth-Heinemann, Oxford (2000).
J. Wloka, Partial diflerential equations. Cambridge University Press (1987).