A Finite-Volume Scheme for Fractional Diffusion on Bounded Domains

TL;DR

This paper introduces a finite-volume scheme for fractional diffusion on bounded domains, enabling accurate numerical solutions and boundary condition implementation for fractional Laplacian operators.

Contribution

It presents a novel fractional Laplacian formulation suitable for finite-volume schemes with no-flux boundary conditions and provides a comprehensive numerical analysis.

Findings

The scheme accurately captures fractional Laplacian effects.

Numerical solutions match analytical benchmarks.

Properties of stationary states and asymptotics are explored.

Abstract

We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the well-posedness theory for the fractional heat equation. We also develop a numerical scheme, which correctly captures the action of the fractional Laplacian and its anomalous diffusion effect. We benchmark numerical solutions for the L\'evy-Fokker-Planck equation against known analytical solutions. We conclude by numerically exploring properties of these equations with respect to their stationary states and long-time asymptotics.