On a structure-preserving numerical method for fractional Fokker-Planck equations

TL;DR

This paper develops and analyzes a numerical scheme for fractional Fokker-Planck equations that preserves key physical properties and demonstrates exponential stability and convergence through theoretical analysis and numerical tests.

Contribution

The paper introduces a novel structure-preserving numerical scheme for fractional Fokker-Planck equations, including new discrete functional analysis tools.

Findings

Scheme preserves mass and heavy-tailed equilibria

Proves exponential stability and convergence

Numerical simulations confirm theoretical results

Abstract

In this paper, we introduce and analyse numerical schemes for the homogeneous and the kinetic L\'evy-Fokker-Planck equation. The discretizations are designed to preserve the main features of the continuous model such as conservation of mass, heavy-tailed equilibrium and (hypo)coercivity properties. We perform a thorough analysis of the numerical scheme and show exponential stability and convergence of the scheme. Along the way, we introduce new tools of discrete functional analysis, such as discrete nonlocal Poincar\'e and interpolation inequalities adapted to fractional diffusion. Our theoretical findings are illustrated and complemented with numerical simulations.