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Numerische Mathematik

, Volume 118, Issue 2, pp 329–366 | Cite as

Analysis of a new class of forward semi-Lagrangian schemes for the 1D Vlasov Poisson equations

  • Thomas Respaud
  • Eric SonnendrückerEmail author
Article

Abstract

The Vlasov equation is a kinetic model describing the evolution of a plasma which is a globally neutral gas of charged particles. It is self-consistently coupled with Poisson’s equation, which rules the evolution of the electric field. In this paper, we introduce a new class of forward semi-Lagrangian schemes for the Vlasov–Poisson system based on a Cauchy Kovalevsky (CK) procedure for the numerical solution of the characteristic curves. Exact conservation properties of the first moments of the distribution function for the schemes are derived and a convergence study is performed that applies as well for the CK scheme as for a more classical Verlet scheme. A L 1 convergence of the schemes will be proved. Error estimates [in \({O\left(\Delta{t}^2+h^2 + \frac{h^2}{\Delta{t}}\right)}\) for Verlet] are obtained, where Δt and h = max(Δx, Δv) are the discretization parameters.

Mathematics Subject Classification (2000)

65M12 65M25 

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References

  1. 1.
    Besse N., Mehrenberger M.: Convergence of classes of high order semi-Lagrangian schemes for the Vlasov–Poisson system. Math. Comput. 77, 93–123 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bostan M., Crouseilles N.: Convergence of a semi-Lagrangian scheme for the reduced Vlasov–Maxwell system for laser-plasma interaction. Numer. Math. 112, 169–195 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bouchut F., Golse F., Pulvirenti M.: Kinetic equations and asymptotic theory. In: Ciarlet, P.G., Lions, P.L. (eds) Series in Applied Math, Gauthier Villars, Paris (2008)Google Scholar
  4. 4.
    Birdsall C.K., Langdon A.B.: Plasma Physics via Computer Simulation. Institute of Physics Publishing, Bristol/Philadelphia (1991)CrossRefGoogle Scholar
  5. 5.
    Carillo J.-A., Vecil F.: Non oscillatory interpolation methods applied to Vlasov–based models. SIAM J. Sci. Comput. 29, 1179–1206 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cheng C.Z., Knorr G.: The integration of the Vlasov equation in configuration space. J. Comput. Phys 22, 330–3351 (1976)CrossRefGoogle Scholar
  7. 7.
    Crouseilles N., Respaud T., Sonnendrücker E.: A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Comput. Phys. Comm. 180(10), 1730–1745 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cottet G.-H., Raviart P.-A.: Particle methods for the one-dimensional Vlasov–Poisson equations. SIAM J. Numer. Anal. 21, 52–75 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Després B.: Finite volume transport Schemes. Numerische Mathematik 108, 529–556 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Filbet F., Sonnendrücker E., Bertrand P.: Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172, 166–187 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Filbet F., Sonnendrücker E.: Comparison of Eulerian Vlasov solvers. Comput. Phys. Comm. 151, 247–266 (2003)CrossRefGoogle Scholar
  12. 12.
    Glassey R.T.: The Cauchy problem in kinetic theory. SIAM, Philadelphia (1996)zbMATHGoogle Scholar
  13. 13.
    Grandgirard V., Brunetti M., Bertrand P., Besse N., Garbet X., Ghendrih P., Manfredi G., Sarrazin Y., Sauter O., Sonnendrücker E., Vaclavik J., Villard L.: A drift-kinetic semi-Lagrangian 4D code for ion turbulence simulation. J. Comput. Phys. 217, 395–423 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Reich S.: An explicit and conservative remapping strategy for semi-Lagrangian advection. Atmos. Sci. Lett. 8, 58–63 (2007)CrossRefGoogle Scholar
  15. 15.
    Staniforth A., Coté J.: Semi-Lagrangian integration schemes for atmospheric models—a review. Mon. Weather Rev. 119, 2206–2223 (1991)CrossRefGoogle Scholar
  16. 16.
    Sonnendrücker E., Roche J., Bertrand P., Ghizzo A.: The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149, 201–220 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Zerroukat M., Wood N., Staniforth A.: A monotonic and positive-definite filter for a Semi-Lagrangian Inherently Conserving and Efficient (SLICE) scheme. Q.J.R. Meteorol. Soc. 131, 2923–2936 (2005)CrossRefGoogle Scholar
  18. 18.
    Zerroukat M., Wood N., Staniforth A.: The Parabolic Spline Method (PSM) for conservative transport problems. Int. J. Numer. Methods Fluids 51, 1297–1318 (2006)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.IRMAUniversité de Strasbourg and INRIA-Nancy-Grand Est, CALVI Project-TeamStrasbourgFrance

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