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

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.

Multiple point evaluation on combined tensor product supports

“…These can be located anywhere, if general flow fields are admitted. The semi-Lagrangian approach in combination with adaptive sparse grid spaces offers a promising numerical technique for moderately high-dimensional boundary value problems arising in areas as diverse as optimal control [4] and kinetic equations [2]. This has motivated the present article.…”

Précision d'un schéma adaptatif semi-lagrangien pour l'équation de Vlasov

“…Ceci en garantissant d'une part la précision des solutions, d'autre part l'optimalité des maillages ainsi construits, du moins dans une certaine mesure. D'abord proposé et analysé dans l'article [3], puis décrit de façon plus rapide dans la note [5], ce schéma aégalement fait l'objet d'une présentation détaillée dans ma thèse [4]. C'est essentiellement l'articulation de cette dernière que je suivrai ici, en considérant un problème de transport "abstrait" de type Vlasov dont j'énoncerai les principales propriétés.…”

“…Moreover, this evolution is done in a way that is in some sense optimal, since on an analytical perspective, the accuracy of the solutions is established, together with a (still incomplete) bound on the complexity of the meshes. First proposed and analyzed in the article [3], then roughly described in a previous proceeding [5], this scheme was also given a detailed presentation in my PhD dissertation [4], in french. I shall here follow the latter and consider some "abstract", Vlasov-type problem which properties will first be recalled.…”