Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-13T05:07:59.137Z Has data issue: false hasContentIssue false
  • English
  • Français

Degenerate Anisotropic Elliptic Problems and Magnetized Plasma Simulations

Published online by Cambridge University Press:  20 August 2015

Stéphane Brull ,
Pierre Degond  and
Fabrice Deluzet
Stéphane Brull*
Affiliation:
Institut de Mathématiques de Bordeaux UMR 5251, Equipe Mathématiques Appliquées de Bordeaux (MAB) Université Bordeaux 1 351, cours de la Libération-33405 TALENCE cedex, France
Pierre Degond
Affiliation:
Université de Toulouse; UPS, INSA, UT1, UTM; Institut de Mathématiques de Toulouse; F-31062 Toulouse, France CNRS; Institut de Mathématiques de Toulouse UMR 5219, F-31062 Toulouse, France
Fabrice Deluzet
Affiliation:
Université de Toulouse; UPS, INSA, UT1, UTM; Institut de Mathématiques de Toulouse; F-31062 Toulouse, France CNRS; Institut de Mathématiques de Toulouse UMR 5219, F-31062 Toulouse, France
*
Corresponding author.Email:stephane.brull@math.u-bordeaux1.fr
Get access

Abstract

This paper is devoted to the numerical approximation of a degenerate anisotropic elliptic problem. The numerical method is designed for arbitrary space-dependent anisotropy directions and does not require any specially adapted coordinate system. It is also designed to be equally accurate in the strongly and the mildly anisotropic cases. The method is applied to the Euler-Lorentz system, in the drift-fluid limit. This system provides a model for magnetized plasmas.

Keywords

65N0665N1265M0665M1276W0576X0576N17Anisotropic elliptic problemasymptotic-preserving schemeLorentz forcelarge magnetic fieldlow-Mach numberdrift-fluid limit
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 1 , January 2012 , pp. 147 - 178
Copyright
Copyright © Global Science Press Limited 2012

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

[1]Beer, M., Cowley, S. and Hammett, G., Field-aligned coordinates for nonlinear simulations of tokamak turbulence, Phys. Plasmas, 2(7) (1995), 2687–2700.Google Scholar
[2]Beer, M. A. and Hammett, G. W., Toroidal gyrofluid equations for simulations of tokamak turbulence, Phys. Plasmas, 3 (1996), 4046–4064.CrossRefGoogle Scholar
[3]Belaouar, R., Crouseilles, N., Degond, P. and Sonnendrcker, E., An asymptotically stable semi-lagrangian scheme in the quasi-neutral limit, J. Sci. Comput., (2009), 341–365.Google Scholar
[4]Besse, C., Deluzet, F., Negulescu, C. and Yang, C., Three dimensional simulation of ionospheric plasma disturbances, in preparation.Google Scholar
[5]Boozer, A. H., Establishment of magnetic coordinates for a given magnetic field, Phys. Fluids, 25(3) (1982), 520–521.CrossRefGoogle Scholar
[6]Buet, C., Cordier, S., Lucquin-Desreux, B. and Mancini, S., Diffusion limit of the Lorentz model: Asymptotic Preserving schemes, ESAIM: M2AN, 36 (2002), 631–655.CrossRefGoogle Scholar
[7]Buet, C. and Despres, B., Asymptotic Preserving and positive schemes for radiation hydrodynamics, J. Comput. Phys., 215 (2006), 717–740.Google Scholar
[8]Carrillo, J. A., Goudon, T. and Lafitte, P., Simulation of fluid and particles flows: Asymptotic Preserving schemes for bubbling and flowing regimes, J. Comput. Phys., 227 (2008), 7929– 7951.Google Scholar
[9]Crispel, P., Degond, P. and Vignal, M.-H., An Asymptotic Preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit, J. Comput. Phys., 223(1) (2007), 208–234.CrossRefGoogle Scholar
[10]Degond, P., Deluzet, F., Lozinski, A., Narski, J. and Negulescu, C., Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations, arXiv:1008.3405v1, 2010.Google Scholar
[11]Degond, P., Deluzet, F., Navoret, L., Sun, A.-B. and Vignal, M.-H., Asymptotic-Preserving Particle-In-Cell method for the Vlasov-Poisson system near quasineutrality, J. Comput. Phys., 229(16) (2010), 5630–5652.Google Scholar
[12]Degond, P., Deluzet, F. and Negulescu, C., An Asymptotic Preserving scheme for strongly anisotropic elliptic problems, Multiscale Model. Sim., 8(2) (2010), 645–666.Google Scholar
[13]Degond, P., Deluzet, F., Sangam, A. and Vignal, M.-H., An Asymptotic-Preserving scheme for the Euler equations in a strong magnetic field, J. Comput. Phys., 228(10) (2009), 3540–3558.Google Scholar
[14]Degond, P., Liu, H., Savelief, D. and Vignal, M. H., Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit, submitted.Google Scholar
[15]Degond, P., Liu, J.-G. and Vignal, M.-H., Analysis of an Asymptotic Preserving scheme for the Euler-Poisson system in the quasineutral limit, SIAM J. Numer. Anal., 46(3) (2008), 1298–1322.Google Scholar
[16]Degond, P. and Tang, M., All speed scheme for the low Mach number limit of the isentropic Euler equation, Commun. Comput. Phys., to appear.Google Scholar
[17]D’haeseleer, W.D., Hitchon, W. N. G., Callen, J. D. and Shohet, J. L., Flux Coordinates and Magnetic Field Structure: A Guide to a Fundamental Tool of Plasma Theory, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991.Google Scholar
[18]Dimits, A. M., Fluid simulations of tokamak turbulence in quasiballooning coordinates, Phys. Rev. E, 48(5) (1993), 4070–4079.CrossRefGoogle ScholarPubMed
[19]Dorland, W. and Hammett, G., Gyrofluid turbulence models with kinetic effects, Phys. Fluids B, 5(3) (1993), 812–835.CrossRefGoogle Scholar
[20]Filbet, F. and Jin, S., A class of Asymptotic Preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., in press.Google Scholar
[21]Giraud, L. and Tuminaro, R. S., Schur complement preconditioners for anisotropic problems, IMA J. Numer. Anal., 19(1) (1999), 1–18.Google Scholar
[22]Grandgirard, V., Sarazin, Y., Garbet, X., Dif-Pradalier, G., Ghendrih, P., Crouseilles, N., Latu, G., Sonnendrücker, E., Besse, N. and Bertrand, P., Computing ITG turbulence with a full-f semi-lagrangian code, Commun. Nonlinear Sci. Numer. Simulat., 13(1) (2008), 81–87.Google Scholar
[23]Hamada, S., Hydromagnetic equilibria and their proper coordinates, Nucl. Fusion, 2 (1962), 23–37.Google Scholar
[24]Hammett, G. W., Beer, M. A., Dorland, W., Cowley, S. C. and Smith, S. A., Developments in the gyrofluid approach to tokamak turbulence simulations, Plasma Phys. Contr. Fusion, 35(8) (1993), 973.Google Scholar
[25]Jin, S., Efficient Asymptotic-Preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21(2) (1999), 441–454.Google Scholar
[26]Kaveeva, E. and Rozhansky, V., Poloidal and toroidal flows in tokamak plasma near magnetic islands, Tech. Phys. Lett., 30(7) (2004), 538–540.Google Scholar
[27]Khoromskij, B. N. and Wittum, G., Robust Schur complement method for strongly anisotropic elliptic equations, Numer. Linear Algebra Appl., 6(8) (1999), 621–653.Google Scholar
[28]Klar, A., An Asymptotic Preserving numerical scheme for kinetic equations in the low Mach number limit, SIAM J. Numer. Anal., 36 (1999), 1507–1527.CrossRefGoogle Scholar
[29]Lemou, M. and Mieussens, L., A new Asymptotic Preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 31 (2008), 334–368.Google Scholar
[30]Leveque, R. J., Finite Volume Method for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.Google Scholar
[31]Llorente, I. and Melson, N., Robust multigrid smoothers for three dimensional elliptic equations with strong anisotropies, ICASE Technical Report: TR-98-37, 1998.Google Scholar
[32]McClarren, R. G. and Lowrie, B., The effects of slope limiting on Asymptotic-Preserving numerical methods for hyperbolic conservation laws, J. Comput. Phys., 227 (2008), 9711–9726.Google Scholar
[33]Melenk, J. M., hp-finite element methods for singular perturbations, Volume 1796 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2002.Google Scholar
[34]Miyamoto, K., Controlled Fusion and Plasma Physics, Chapman & Hall, 2007.Google Scholar
[35]Ottaviani, M. A., An alternative approach to field-aligned coordinates for plasma turbulence simulations, arXiv:1002.0748, 2010.Google Scholar
[36]Stern, D., Geomagnetic Euler potentials, J. Geophys. Res., 72(15) (1967), 3995–4005.CrossRefGoogle Scholar