An asymptotic preserving scheme for L\'{e}vy-Fokker-Planck equation with fractional diffusion limit
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.
Keywords
Cite
@article{arxiv.2103.08848,
title = {An asymptotic preserving scheme for L\'{e}vy-Fokker-Planck equation with fractional diffusion limit},
author = {Wuzhe Xu and Li Wang},
journal= {arXiv preprint arXiv:2103.08848},
year = {2021}
}