Variational Asymptotic Preserving Scheme for the Vlasov-Poisson-Fokker-Planck System

TL;DR

This paper introduces a novel variational asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck system, combining implicit-explicit methods with Wasserstein gradient flows to ensure positivity, asymptotic preservation, and high-dimensional efficiency.

Contribution

It develops a new implicit solver using a variational approach as a Wasserstein gradient flow, enabling positivity, asymptotic preservation, and parallelization for high-dimensional kinetic equations.

Findings

The scheme is positivity-preserving and asymptotic-preserving.

The implicit solver converges uniformly across scales.

Numerical examples validate the scheme's effectiveness.

Abstract

We design a variational asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck system with the high field scaling, which describes the Brownian motion of a large system of particles in a surrounding bath. Our scheme builds on an implicit-explicit framework, wherein the stiff terms coming from the collision and field effects are solved implicitly while the convection terms are solved explicitly. To treat the implicit part, we propose a variational approach by viewing it as a Wasserstein gradient flow of the relative entropy, and solve it via a proximal quasi-Newton method. In so doing we get positivity and asymptotic preservation for free. The method is also massively parallelizable and thus suitable for high dimensional problems. We further show that the convergence of our implicit solver is uniform across different scales. A suite of numerical examples are presented at the end to validate the performance of the proposed scheme.