Diffusion is one of the most common phenomenon in natural sciences and large part of applied mathematics have been interested in the tools to model it. Trying to study different types of diffusions, the mathematical ways to describe them and the numerical methods to simulate them is an appealing challenge, giving a wide range of applications. The aim of our work is the design of a finite-volume numerical scheme to model non-local diffusion given by the fractional Laplacian and to build numerical solutions for the Lévy-Fokker-Planck equation that involves it. Numerical methods for fractional diffusion have been indeed developed during the last few years and large part of the literature has been focused on finite element methods. Few results have been rather proposed for different techniques such as finite volumes.
We propose a new fractional Laplacian for bounded domains, which is expressed as a conservation law. This new approach is therefore particularly suitable for a finite volumes scheme and allows us also to prescribe no-flux boundary conditions explicitly. We enforce our new definition with a well-posedness theory for some cases to then capture with a good level of approximation the action of fractional Laplacian and its anomalous diffusion effect with our numerical scheme. The numerical solutions we get for the Lévy-Fokker-Planck equation resemble in fact the known analytical predictions and allow us to numerically explore properties of this equation and compute stationary states and long-time asymptotics.