Numerical Methods for Fractional Diffusion

TL;DR

This paper introduces three numerical schemes for fractional diffusion, each based on different definitions, analyzing their advantages, limitations, and performance through error estimates and numerical experiments.

Contribution

It presents novel numerical methods for fractional diffusion based on spectral, integral, and Dunford-Taylor formulations, with comprehensive analysis and comparison.

Findings

All three methods effectively approximate fractional diffusion.

Error estimates demonstrate the accuracy of each scheme.

Numerical experiments validate the theoretical analysis.

Abstract

We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford-Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.