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Journal of Scientific Computing

, Volume 41, Issue 3, pp 341–365 | Cite as

An Asymptotically Stable Semi-Lagrangian scheme in the Quasi-neutral Limit

  • R. Belaouar
  • N. Crouseilles
  • P. Degond
  • E. Sonnendrücker
Article

Abstract

This paper deals with the numerical simulations of the Vlasov-Poisson equation using a phase space grid in the quasi-neutral regime. In this limit, explicit numerical schemes suffer from numerical constraints related to the small Debye length and large plasma frequency. Here, we propose a semi-Lagrangian scheme for the Vlasov-Poisson model in the quasi-neutral limit. The main ingredient relies on a reformulation of the Poisson equation derived in (Crispel et al. in C. R. Acad. Sci. Paris, Ser. I 341:341–346, 2005) which enables asymptotically stable simulations. This scheme has a comparable numerical cost per time step to that of an explicit scheme. Moreover, it is not constrained by a restriction on the size of the time and length step when the Debye length and plasma period go to zero. A stability analysis and numerical simulations confirm this statement.

Keywords

Vlasov equation Quasi-neutral limit Semi-Lagrangian method Asymptotic preserving scheme 

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References

  1. 1.
    Birdsall, C.K., Langdon, A.B.: Plasma Physics via Computer Simulation. IOP Publishing, Bristol (1991) CrossRefGoogle Scholar
  2. 2.
    Brenier, Y.: Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Commun. Part. Differ. Equ. 25, 737–754 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cohen, B.I., Langdon, A.B., Friedman, A.: Implicit time integration for plasma simulations. J. Comput. Phys. 46, 15 (1982) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Crispel, P., Degond, P., Vignal, M.-H.: An asymptotically stable discretization for the Euler-Poisson system in the quasineutral limit. C.R. Acad. Sci. Paris, Ser. I 341, 341–346 (2005) MathSciNetGoogle Scholar
  5. 5.
    Crispel, P., Degond, P., Vignal, M.-H.: An asymptotically preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit. J. Comput. Phys. 203, 208–234 (2007) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Crouseilles, N., Filbet, F.: Numerical approximation of collisional plasma by high order methods. J. Comput. Phys. 201(2), 546–572 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Crouseilles, N., Latu, G., Sonnendrücker, E.: Hermite spline interpolation on patches for parallelly solving the Vlasov-Poisson equation. Int. J. Appl. Math. Comput. Sci. 17(3), 101–115 (2007) CrossRefGoogle Scholar
  8. 8.
    Degond, P., Parzani, C., Vignal, M.-H.: Plasma expansion in vacuum: modeling the breakdown of quasineutrality. SIAM Multiscale Model. Simul. 2, 158 (2003) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Degond, P., Deluzet, F., Navoret, L.: An asymptotically stable Particle-In-cell (PIC) scheme for collisionless plasma simulations near quasineutrality. C.R. Acad. Sci. Paris, Ser. I 343, 613–618 (2006) zbMATHMathSciNetGoogle Scholar
  10. 10.
    Degond, P., Liu, J.G., Vignal, M.-H.: Analysis of an asymptotic preserving scheme for the Euler-Poisson system in the quasineutral limit. SIAM J. Numer. Anal. 46, 1298–1322 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Delcroix, J.-L., Bers, A.: Physique des Plasmas, vols. 1, 2. Inter Editions/Editions du CNRS, Paris (1994) Google Scholar
  12. 12.
    Duclous, R., Dubroca, B., Filbet, F.: Analysis of a high order finite volume scheme for the Vlasov-Poisson system. http://hal.archives-ouvertes.fr/hal-00287630/fr
  13. 13.
    Fabre, S.: Stability analysis of the Euler-Poisson equations. J. Comput. Phys. 101, 445 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Filbet, F., Sonnendrücker, E.: Comparison of Eulerian Vlasov solvers. Comput. Phys. Commun. 151, 247–266 (2003) CrossRefGoogle Scholar
  15. 15.
    Filbet, F., Sonnendrücker, E., Bertrand, P.: Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172, 166–187 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Fried, B.D., Comte, S.D.: The Plasma Dispersion Function. Academic Press, New York (1961) Google Scholar
  17. 17.
    Ha, S.Y., Slemrod, M.: Global existence of plasma ion sheaths and their dynamics. Commun. Math. Phys. 238, 149 (2003) zbMATHMathSciNetGoogle Scholar
  18. 18.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations. Series in Computational Mathematics, vol. 31. Springer, Berlin (2002) zbMATHGoogle Scholar
  19. 19.
    Hewett, D.W., Nielson, C.W.: A multidimensional quasineutral plasma simulation model. J. Comput. Phys. 72, 121 (1987) zbMATHCrossRefGoogle Scholar
  20. 20.
    Hockney, R.W., Eastwood, J.W.: Computer Simulation Using Particles. IOP Publishing, Bristol (1998) Google Scholar
  21. 21.
    Langdon, A.B., Cohen, B.I., Friedman, A.: Direct implicit large time-step particle simulation of plasmas. J. Comput. Phys. 51, 107 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Mankofsky, A., Sudan, R.N., Denavit, J.: Hybrid simulation of ion beams in background plasma. J. Comput. Phys. 51, 484 (1983) CrossRefGoogle Scholar
  23. 23.
    Masson, R.J.: Implicit moment particle simulations of plasmas. J. Comput. Phys. 41, 233 (1981) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Masson, R.J.: Implicit moment PIC-hybrid simulation of collisional plasmas. J. Comput. Phys. 51, 484 (1983) CrossRefGoogle Scholar
  25. 25.
    Nakamura, T., Yabe, T.: Cubic interpolated propagation scheme for solving the hyper-dimensional Vlasov-Poisson equation in phase space. Comput. Phys. Commun. 120, 122–154 (1999) zbMATHCrossRefGoogle Scholar
  26. 26.
    Rambo, P.W.: Finite-grid instability in quasineutral hybrid simulations. J. Comput. Phys. 118, 152 (1995) zbMATHCrossRefGoogle Scholar
  27. 27.
    Shoucri, M.: Nonlinear evolution of the bump-on-tail instability. Phys. Fluids 22, 2038–2039 (1979) CrossRefGoogle Scholar
  28. 28.
    Sonnendrücker, E., Roche, J., Bertrand, P., Ghizzo, A.: The semi-Lagrangian method for the numerical resolution of the Vlasov equations. J. Comput. Phys. 149, 201–220 (1999) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • R. Belaouar
    • 1
  • N. Crouseilles
    • 2
  • P. Degond
    • 3
    • 4
  • E. Sonnendrücker
    • 2
  1. 1.Centre de Mathématiques Appliquées (UMR 7641)Ecole PolytechniquePalaiseauFrance
  2. 2.INRIA Nancy-Grand-EST (CALVI Project), and IRMA-Université de Strasbourg and CNRSStrasbourg CedexFrance
  3. 3.1-Université de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathématiques de ToulouseToulouseFrance
  4. 4.2-CNRS, Institut de Mathématiques de Toulouse UMR 5219ToulouseFrance

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