A finite-volume scheme for fractional diffusion on bounded domains

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 Lévy–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.