A Higher Order Resolvent-positive Finite Difference Approximation for Fractional Derivatives

TL;DR

This paper introduces a higher-order finite difference scheme for fractional derivatives that maintains positivity and order, outperforming the traditional Grunwald scheme especially for skewed fractional heat equations.

Contribution

A novel finite difference approximation of order α for fractional derivatives that preserves positivity and maintains its order in skewed fractional heat equations.

Findings

Preserves positivity of solutions.

Maintains order α for skewed fractional heat equations.

Outperforms Grunwald scheme in convergence rate.

Abstract

We develop a finite difference approximation of order $\alpha$ for the $\alpha$-fractional derivative. The weights of the approximation scheme have the same rate-matrix type properties as the popular Gr\"unwald scheme. In particular, approximate solutions to fractional diffusion equations preserve positivity. Furthermore, for the approximation of the solution to the skewed fractional heat equation on a bounded domain the new approximation scheme keeps its order $\alpha$ whereas the order of the Gr\"unwald scheme reduces to order $\alpha-1$, contradicting the convergence rate results by Meerschaert and Tadjeran.