Asymptotic preserving schemes for highly oscillatory kinetic equation

TL;DR

This paper develops an Asymptotic Preserving numerical scheme for the Vlasov-Poisson equation with rapidly oscillating external fields, allowing efficient simulation across oscillatory regimes without time step refinement.

Contribution

It introduces a novel double-scale reformulation and numerical scheme that works uniformly in both oscillatory and non-oscillatory regimes, without deriving asymptotic models.

Findings

Scheme accurately captures highly oscillatory behavior

No time step refinement needed for simulations

Works in both oscillatory and non-oscillatory regimes

Abstract

This work is devoted to the numerical simulation of a Vlasov-Poisson model describing a charged particle beam under the action of a rapidly oscillating external electric field. We construct an Asymptotic Preserving numerical scheme for this kinetic equation in the highly oscillatory limit. This scheme enables to simulate the problem without using any time step refinement technique. Moreover, since our numerical method is not based on the derivation of the simulation of asymptotic models, it works in the regime where the solution does not oscillate rapidly, and in the highly oscillatory regime as well. Our method is based on a "double-scale" reformulation of the initial equation, with the introduction of an additional periodic variable.