Uniform error estimate of an asymptotic preserving scheme for the L\'{e}vy-Fokker-Planck equation

TL;DR

This paper proves a uniform error estimate for an asymptotic preserving scheme applied to the Lévy-Fokker-Planck equation, addressing challenges from tail decay and parameter interplay, valid across all dimensions and fractional powers.

Contribution

It introduces a novel approach to obtain uniform error bounds for the scheme, handling large and small scaling regimes effectively.

Findings

Established a uniform-in-scaling error estimate for the scheme.

Proved strong convergence of LFP to fractional diffusion with explicit rate.

Extended AP estimates to cases with unbounded uniform bounds.

Abstract

We establish a uniform-in-scaling error estimate for the asymptotic preserving scheme proposed in \cite{XW21} for the L\'evy-Fokker-Planck (LFP) equation. The main difficulties stem from not only the interplay between the scaling and numerical parameters but also the slow decay of the tail of the equilibrium state. We tackle these problems by separating the parameter domain according to the relative size of the scaling $\epsilon$: in the regime where $\epsilon$ is large, we design a weighted norm to mitigate the issue caused by the fat tail, while in the regime where $\epsilon$ is small, we prove a strong convergence of LFP towards its fractional diffusion limit with an explicit convergence rate. This method extends the traditional AP estimates to cases where uniform bounds are unavailable. Our result applies to any dimension and to the whole span of the fractional power.