Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov-Poisson system


Authors: Nicolas Besse and Michel Mehrenberger
Journal: Math. Comp. 77 (2008), 93-123
MSC (2000): Primary 65M12
DOI: https://doi.org/10.1090/S0025-5718-07-01912-6
Published electronically: June 18, 2007
MathSciNet review: 2353945
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we present some classes of high-order semi-Lagran- gian schemes for solving the periodic one-dimensional Vlasov-Poisson system in phase-space on uniform grids. We prove that the distribution function $ f(t,x,v)$ and the electric field $ E(t,x)$ converge in the $ L^2$ norm with a rate of

$\displaystyle \mathcal{O}\left(\Delta t^2 +h^{m+1}+ \frac{h^{m+1}}{\Delta t}\right),$

where $ m$ is the degree of the polynomial reconstruction, and $ \Delta t$ and $ h$ are respectively the time and the phase-space discretization parameters.


References [Enhancements On Off] (What's this?)

  • 1. B. J. C. Baxter, N. Sivakumar, On shifted cardinal interpolation by Gaussians and multiquadrics, J. Approx. Theory, 87 (1996), 36-59. MR 1410611 (97e:41015)
  • 2. M. L. Begue, A. Ghizzo, P. Bertrand, Two-dimensional Vlasov simulation of Raman scattering and plasma beatwave acceleration on parallel computers, J. Comput. Phys., 151 (1999), 458-478.
  • 3. R. Bermejo, Analysis of an algorithm for the Galerkin-characteristic method, Numer. Math., 60 (1991), 163-194. MR 1133578 (93e:65132)
  • 4. R. Bermejo, Analysis of a class of quasi-monotone and conservative semi-Lagrangian advection schemes, Numer. Math., 87 (2001), 597-623. MR 1815727 (2001j:65130)
  • 5. N. Besse, Etude mathématique et numérique de l'équation de Vlasov non linéaire sur des maillages non structurés de l'espace des phases Ph.D. thesis of Institut de Recherche Mathématique Avancée, IRMA, Université Louis Pasteur, Strasbourg, 2003.
  • 6. N. Besse, Convergence of a semi-Lagrangian scheme for the one-dimensional Vlasov-Poisson system, SIAM, J. Numer. Anal., 42 (2004), 350-382. MR 2051070 (2005b:65098)
  • 7. N. Besse, Convergence of a high-order semi-Lagrangian scheme with propagation of gradients for the Vlasov-Poisson system, submitted.
  • 8. N. Besse, E. Sonnendrücker, Semi-Lagrangian schemes for the Vlasov equation on an unstructured mesh of phase space, J. Comput. Phys., 191 (2) (2003), 341-376. MR 2016914 (2004j:82043)
  • 9. N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker, Adaptive numerical method for the Vlasov equation based on a multiresolution analysis, In F. Brezzi, A. Buffa, S. Escorsaro, and A. Murli editors, Numerical Mathematics and Advanced Applications ENUMATH 01, 437-446, Springer 2001.
  • 10. F. Bouchut, F. Golse, M. Pulvirenti, Kinetic equations and asymptotic theory, Series in Applied Mathematics, P.G. Ciarlet and P.-L. Lions (Eds.) Gauthier-Villars (2000). MR 2065070 (2005d:82102)
  • 11. Carl de Boor, On the cardinal spline interpolant to $ e^{int}$, SIAM J. Math. Anal, Vol. 7, No. 6. November 1976. MR 0493056 (58:12096)
  • 12. M. Campos Pinto, M. Mehrenberger, Convergence of an adaptive scheme for the one-dimensional Vlasov-Poisson system, submitted.
  • 13. C. Z. Cheng, G. Knorr, The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22 (1976), 330-351.
  • 14. M. Falcone, R. Ferretti, Convergence analysis for a class of high-order semi-Lagrangian advection schemes, SIAM, J. Numer. Anal., 35 (1998), 909-940. MR 1619910 (99c:65164)
  • 15. F. Filbet, E. Sonnendrücker, P. Bertrand, Conservative numerical schemes for the Vlasov equation, J. Comput. Phys., 172 (2001), 166-187. MR 1852326 (2002h:76106)
  • 16. E. Forest, J. Bengtsson, Application of the Yoshida-Ruth techniques to implicit integration and multi-map explicit integration, Phys. Lett. A 158 (1991), pp. 99-101. MR 1124345 (92f:70013)
  • 17. D. Goldman, T.J. Kaper, Nth-order operator splitting schemes and non reversible systems, SIAM J. Numer. Anal., 33 (1996), 349-367. MR 1377257 (97a:65063)
  • 18. R. T. Glassey, The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathematics (SIAM), Philadephia, PA, 1996. MR 1379589 (97i:82070)
  • 19. H. Hong, S. Steinberg, Accuracy and stability of polynomial interpolation schemes for advection equations, preprint, 2001.
  • 20. P. G. Ciarlet, in Handbook of Numerical analysis, Finite element methods (part 1), Vol. II, P. G. Ciarlet and J. L. Lions (Eds.) North-Holland (1991). MR 1115235 (92f:65001)
  • 21. K. Jetter, S. D. Riemenschneider, N. Sivakumar, Schoenberg's exponential Euler spline curves, Proceedings of the Royal Society of Edinburgh, 118A, 21-33, 1991. MR 1113840 (92f:41021)
  • 22. F. J. Narcowich, N. Sivakumar, J. D. Ward, Stability results for scattered-data interpolation on euclidean spheres, Adv. Comput. Math., 8(3) (1998), 137-163. MR 1628253 (2000d:65016)
  • 23. I. J. Schoenberg, Cardinal interpolation and spline functions, Journal of Approximation Theory, 2, (1969), 167-206. MR 0257616 (41:2266)
  • 24. E. Sonnendrücker, J. Roche, P. Bertrand, A. Ghizzo, The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149 (1999), no. 2, 201-220. MR 1672731 (99j:76100)
  • 25. A. Staniforth, J. Cote, Semi-Lagrangian integration schemes for atmospheric models-a review, Monthly Weather Review, 119 (1991), 2206-2223.
  • 26. G. Strang, Trigonometric polynomials and difference methods of maximum accuracy, J. Math. Phys, 41(2) (1962), 147-154.
  • 27. H. Yoshida, Construction of higher order sympletic integrators, Phys. Let. A 150 (1990), 262-268. MR 1078768 (91h:70014)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M12

Retrieve articles in all journals with MSC (2000): 65M12


Additional Information

Nicolas Besse
Affiliation: Institut de Recherche Mathematique Avancée, Université Louis Pasteur - CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
Address at time of publication: IECN UMR CNRS 7502 and LPMIA UMR CNRS 7040, Université Henri Poincaré Nancy I, Boulevard des Aiguillettes, B.P. 239 F-54506, Vandoeuvre-lès-Nancy, Cedex, France
Email: besse@iecn.u-nancy.fr

Michel Mehrenberger
Affiliation: Institut de Recherche Mathematique Avancée, Université Louis Pasteur - CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
Email: mehrenbe@math.u-strasbg.fr

DOI: https://doi.org/10.1090/S0025-5718-07-01912-6
Received by editor(s): March 29, 2005
Received by editor(s) in revised form: May 25, 2005
Published electronically: June 18, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society