An asymptotic preserving scheme for L\'{e}vy-Fokker-Planck equation with fractional diffusion limit

TL;DR

This paper introduces an asymptotic preserving numerical scheme for the Lévy-Fokker-Planck equation with fractional diffusion, effectively handling nonlocality and multiscale challenges to improve accuracy and efficiency.

Contribution

The paper presents a novel micro-macro decomposition-based scheme that preserves the fractional diffusion limit and efficiently addresses the nonlocal and multiscale features of the equation.

Findings

The scheme accurately captures the fractional diffusion limit.

Numerical examples demonstrate high efficiency and precision.

The method effectively manages the nonlocal fractional Laplacian.

Abstract

In this paper, we develop a numerical method for the L\'evy-Fokker-Planck equation with the fractional diffusive scaling. There are two main challenges. One comes from a two-fold nonlocality, that is, the need to apply the fractional Laplacian operator to a power law decay distribution. The other arises from long-time/small mean-free-path scaling, which introduces stiffness to the equation. To resolve the first difficulty, we use a change of variable to convert the unbounded domain into a bounded one and then apply the Chebyshev polynomial based pseudo-spectral method. To treat the multiple scales, we propose an asymptotic preserving scheme based on a novel micro-macro decomposition that uses the structure of the test function in proving the fractional diffusion limit analytically. Finally, the efficiency and accuracy of our scheme are illustrated by a suite of numerical examples.