A finite difference method for space fractional differential equations with variable diffusivity coefficient

TL;DR

This paper introduces the first finite difference scheme for variable-coefficient fractional differential equations with two-sided derivatives, proving its stability, convergence, and demonstrating its effectiveness through numerical tests.

Contribution

It develops and analyzes a novel finite difference method for variable-coefficient fractional equations, bridging a gap in numerical solutions for such nonlocal models.

Findings

The scheme is globally first-order accurate in space.

Existence and uniqueness of solutions are established.

Numerical tests confirm the scheme's effectiveness and accuracy.

Abstract

Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving {\em variable-coefficient} fractional differential equations, with two-sided fractional derivatives, in one-dimensional space. The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided fractional derivative when the right-sided fractional derivative is approximated by two consecutive applications of the first-order backward Euler method. Our finite difference scheme reduces to the standard second-order central difference scheme in the absence of fractional derivatives. The existence and uniqueness of the solution for the proposed scheme are proved, and truncation errors of order $h$ are demonstrated, where $h$ denotes the maximum space step size. The numerical tests illustrate the global $O(h)$ accuracy of our scheme, except for nonsmooth cases which, as expected, have deteriorated convergence rates.