A direct and accurate adaptive semi-Lagrangian scheme for the Vlasov-Poisson equation

Martin Campos Pinto

International Journal of Applied Mathematics and Computer Science (2007)

  • Volume: 17, Issue: 3, page 351-359
  • ISSN: 1641-876X

Abstract

top
This article aims at giving a simplified presentation of a new adaptive semi-Lagrangian scheme for solving the (1+1)-dimensional Vlasov-Poisson system, which was developed in 2005 with Michel Mehrenberger and first described in (Campos Pinto and Mehrenberger, 2007). The main steps of the analysis are also given, which yield the first error estimate for an adaptive scheme in the context of the Vlasov equation. This article focuses on a key feature of our method, which is a new algorithm to transport multiscale meshes along a smooth flow, in a way that can be said optimal in the sense that it satisfies both accuracy and complexity estimates which are likely to lead to optimal convergence rates for the whole numerical scheme. From the regularity analysis of the numerical solution and how it gets transported by the numerical flow, it is shown that the accuracy of our scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step. As a consequence, the numerical solutions are proved to converge in L^∞ towards the exact ones as ε and Δt tend to zero, and in addition to the numerical tests presented in (Campos Pinto and Mehrenberger, 2007), some complexity bounds are established which are likely to prove the optimality of the meshes.

How to cite

top

Campos Pinto, Martin. "A direct and accurate adaptive semi-Lagrangian scheme for the Vlasov-Poisson equation." International Journal of Applied Mathematics and Computer Science 17.3 (2007): 351-359. <http://eudml.org/doc/207842>.

@article{CamposPinto2007,
abstract = {This article aims at giving a simplified presentation of a new adaptive semi-Lagrangian scheme for solving the (1+1)-dimensional Vlasov-Poisson system, which was developed in 2005 with Michel Mehrenberger and first described in (Campos Pinto and Mehrenberger, 2007). The main steps of the analysis are also given, which yield the first error estimate for an adaptive scheme in the context of the Vlasov equation. This article focuses on a key feature of our method, which is a new algorithm to transport multiscale meshes along a smooth flow, in a way that can be said optimal in the sense that it satisfies both accuracy and complexity estimates which are likely to lead to optimal convergence rates for the whole numerical scheme. From the regularity analysis of the numerical solution and how it gets transported by the numerical flow, it is shown that the accuracy of our scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step. As a consequence, the numerical solutions are proved to converge in L^∞ towards the exact ones as ε and Δt tend to zero, and in addition to the numerical tests presented in (Campos Pinto and Mehrenberger, 2007), some complexity bounds are established which are likely to prove the optimality of the meshes.},
author = {Campos Pinto, Martin},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {semi-Lagrangian method; error estimates; optimal transport of adaptive multiscale meshes; fully adaptive scheme; convergence rates; Vlasov-Poisson system},
language = {eng},
number = {3},
pages = {351-359},
title = {A direct and accurate adaptive semi-Lagrangian scheme for the Vlasov-Poisson equation},
url = {http://eudml.org/doc/207842},
volume = {17},
year = {2007},
}

TY - JOUR
AU - Campos Pinto, Martin
TI - A direct and accurate adaptive semi-Lagrangian scheme for the Vlasov-Poisson equation
JO - International Journal of Applied Mathematics and Computer Science
PY - 2007
VL - 17
IS - 3
SP - 351
EP - 359
AB - This article aims at giving a simplified presentation of a new adaptive semi-Lagrangian scheme for solving the (1+1)-dimensional Vlasov-Poisson system, which was developed in 2005 with Michel Mehrenberger and first described in (Campos Pinto and Mehrenberger, 2007). The main steps of the analysis are also given, which yield the first error estimate for an adaptive scheme in the context of the Vlasov equation. This article focuses on a key feature of our method, which is a new algorithm to transport multiscale meshes along a smooth flow, in a way that can be said optimal in the sense that it satisfies both accuracy and complexity estimates which are likely to lead to optimal convergence rates for the whole numerical scheme. From the regularity analysis of the numerical solution and how it gets transported by the numerical flow, it is shown that the accuracy of our scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step. As a consequence, the numerical solutions are proved to converge in L^∞ towards the exact ones as ε and Δt tend to zero, and in addition to the numerical tests presented in (Campos Pinto and Mehrenberger, 2007), some complexity bounds are established which are likely to prove the optimality of the meshes.
LA - eng
KW - semi-Lagrangian method; error estimates; optimal transport of adaptive multiscale meshes; fully adaptive scheme; convergence rates; Vlasov-Poisson system
UR - http://eudml.org/doc/207842
ER -

References

top
  1. Besse N. (2004): Convergence of a semi-Lagrangian scheme for the one-dimensional Vlasov-Poisson system. SIAM Journal on Numerical Analysis, Vol. 42, No. 1, pp. 350-382. Zbl1071.82037
  2. Besse N., Filbet F., Gutnic M., Paun I. and Sonnendrucker E. (2001): An adaptive numerical method for the Vlasov equation based on a multiresolution analysis, In: Numerical Mathematics and Advanced Applications ENUMATH 2001(F. Brezzi, A. Buffa, S. Escorsaro and A. Murli, Eds.). Ischia: Springer, pp. 437-446. Zbl1254.82035
  3. Campos Pinto M. (2005): Developpement et analyse de schemas adaptatifs pour les equations de transport. Ph. D. thesis (in French), Universite Pierre et Marie Curie, Paris. 
  4. Campos Pinto M. (2007): P_1 interpolation in the plane and functions of bounded total curvature. (in preparation). 
  5. Campos Pinto M. and Mehrenberger M. (2005): Adaptive numerical resolution of the Vlasov equation, In: Numerical Methods for Hyperbolic and Kinetic Problems (S. Cordier, T. Goudon, M. Gutnic, E. Sonnendrucker, Eds.). Zurich: European Mathematical Society, Vol. 7, pp. 43-58. Zbl1210.65169
  6. Campos Pinto M. and Mehrenberger M. (2007): Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system. (submitted) Zbl1187.65108
  7. Cheng C. Z. and Knorr G. (1976): The integration of the Vlasov equation in configuration space. Journal of Computational Physics, Vol. 22, pp. 330-351. 
  8. Cohen A., Kaber S. M., Muller S. and Postel M. (2003): Fully adaptive multiresolution finite volume schemes for conservation laws. Mathematics of Computation, Vol. 72, No. 241, pp. 183-225. Zbl1010.65035
  9. Cooper J. and Klimas A. (1980): Boundary value problems for the Vlasov-Maxwell equation in one dimension. Journal of Mathematical Analysis and Applications, Vol. 75, No. 2, pp. 306-329. Zbl0454.35075
  10. Dahmen, W. (1982): Adaptive approximation by multivariate smooth splines. Journal of Approximation Theory, Vol. 36, No. 2, pp. 119-140. Zbl0493.41009
  11. Dahmen W., Gottschlich-Muller B. and Muller S. (2001): Multiresolution schemes for conservation laws. Numerische Mathematik, Vol. 88, No. 3, pp. 399-443. Zbl1001.65104
  12. DeVore, R. (1998): Nonlinear approximation. Acta Numerica, Vol. 7, pp. 51-150 Zbl0931.65007
  13. Glassey R. T. (1996): The Cauchy Problem in Kinetic Theory. Philadelphia, PA: SIAM Zbl0858.76001
  14. Gutnic M., Haefele M., Paun I. and Sonnendrucker E. (2004): Vlasov simulations on an adaptive phase-space grid. Computer Physics Communications, Vol. 164, No. 1-3, pp. 214-219. Zbl1196.76098
  15. Iordanskii S. V. (1964): The Cauchy problem for the kinetic equation of plasma. American Mathematical Society Tranations, Series2, Vol. 35, pp. 351-363. 
  16. Raviart P. -A. (1985): An analysis of particle methods. Lecture Notes in Mathematics, Vol. 1127, Springer, Berlin, pp. 243-324. 
  17. Roussel O., Schneider K., Tsigulin, A. and Bockhorn H. (2003): A conservative fully adaptive multiresolution algorithm for parabolic PDEs. Journal of Computational Physics, Vol. 188, No. 2, pp. 493-523. Zbl1022.65093
  18. Sonnendrucker E., Filbet F., Friedman A., Oudet E. and Vay J. L. (2004): Vlasov simulation of beams with a moving grid. Computer Physics Communications, Vol. 164, pp. 390-395. 
  19. Sonnendrucker E., Roche J., Bertrand P. and Ghizzo A. (1999): The semi-Lagrangian method for the numerical resolution of the Vlasov equation. Journal of Computational Physics, Vol. 149, No. 2, pp. 201-220. Zbl0934.76073
  20. Yserentant H. (1992): Hierarchical bases, In: ICIAM 91: Proceedings of the 2nd International Conference on Industrial and Applied Mathematics (R. E. O'Malley, Ed.). Philadelphia, PA: SIAM, pp. 256-276 

NotesEmbed ?

top

You must be logged in to post comments.