A Separable and Asymptotic-Preserving Dynamical Low-Rank Method for the Vlasov--Poisson--Fokker--Planck System

TL;DR

This paper introduces a novel dynamical low-rank method for the Vlasov--Poisson--Fokker--Planck system that efficiently handles stiff collisions and preserves asymptotic behavior, demonstrated through numerical experiments.

Contribution

It develops a conservative spatial discretization and low-rank IMEX schemes for the VPFP system, with proven asymptotic-preserving properties and improved efficiency.

Findings

Accurate and robust numerical results at modest ranks.

Proof of asymptotic-preserving property for the first-order scheme.

Efficient low-rank projection enabled by factorized discretization.

Abstract

We present a dynamical low-rank (DLR) method for the Vlasov--Poisson--Fokker--Planck (VPFP) system. Our main contributions are two-fold: (i) a conservative spatial discretization of the Fokker--Planck operator that factors into velocity-only and space-only components, enabling efficient low-rank projection, and (ii) a time discretization within the DLR framework that properly handles stiff collisions. We propose both first-order and second-order low-rank IMEX schemes. For the first-order scheme, we prove an asymptotic-preserving (AP) property when the field fluctuation is small. Numerical experiments demonstrate accuracy, robustness, and AP property at modest ranks.