<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml"><p>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 <italic>optimal</italic> 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 <italic>L</italic><sup>∞</sup> towards the exact ones as ε and δ<italic>t</italic> 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.</p></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.

A Direct and Accurate Adaptive Semi-Lagrangian Scheme for the Vlasov-Poisson Equation

Université Louis Pasteur, IRMA, CNRS UMR 7501 7, rue René Descartes, 67084 Strasbourg Cedex, France

International Journal of Applied Mathematics and Computer Science

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...

A Direct and Accurate Adaptive Semi-Lagrangian Scheme for the Vlasov-Poisson Equation

Published Online: Oct 11, 2007

Page range: 351 - 359

DOI: https://doi.org/10.2478/v10006-007-0029-9

Keywords
fully adaptive scheme, semi-Lagrangian method, Vlasov-Poisson system, error estimates, convergence rates, optimal transport of adaptive multiscale meshes

This content is open access.

A Direct and Accurate Adaptive Semi-Lagrangian Scheme for the Vlasov-Poisson Equation

Published Online: Oct 11, 2007

Page range: 351 - 359

DOI: https://doi.org/10.2478/v10006-007-0029-9

Keywordsfully adaptive scheme, semi-Lagrangian method, Vlasov-Poisson system, error estimates, convergence rates, optimal transport of adaptive multiscale meshes

This content is open access.

Keywords
fully adaptive scheme, semi-Lagrangian method, Vlasov-Poisson system, error estimates, convergence rates, optimal transport of adaptive multiscale meshes