Communications in Mathematical Sciences

Volume 21 (2023)

Number 1

An asymptotic preserving scheme for Lévy-Fokker-Planck equation with fractional diffusion limit

Pages: 1 – 23



Wuzhe Xu (School of Mathematics, University of Minnesota, Minneapolis, Mn., U.S.A.)

Li Wang (School of Mathematics, University of Minnesota, Minneapolis, Mn., U.S.A.)


In this paper, we develop a numerical method for the Lévy–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 into 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.


asymptotic preserving, fractional Laplacian, Lévy–Fokker–Planck, micro-macro decomposition

2010 Mathematics Subject Classification

65M70, 82C40, 82D99

This work is partially supported by NSF grant DMS-1846854.

Received 29 March 2021

Received revised 11 January 2022

Accepted 23 March 2022

Published 27 December 2022