Finite difference/spectral approximations for the two-dimensional time Caputo-Fabrizio fractional diffusion equation

TL;DR

This paper develops a stable, high-order numerical scheme combining finite difference and spectral methods to efficiently solve the two-dimensional time Caputo-Fabrizio fractional diffusion equation, with proven stability and verified by numerical experiments.

Contribution

It introduces the first stability proof for a third-order scheme solving the Caputo-Fabrizio fractional diffusion equation, combining finite difference and spectral methods.

Findings

The scheme is unconditionally stable.

Global truncation error is $ ext{O}( au^3+N^{-m})$.

Numerical experiments confirm theoretical results.

Abstract

The main contribution of this work is to construct and analyze stable and high order schemes to efficiently solve the two-dimensional time Caputo-Fabrizio fractional diffusion equation. Based on a third-order finite difference method in time and spectral methods in space, the proposed scheme is unconditionally stable and has the global truncation error $\mathcal{O}(\tau^3+N^{-m})$, where $\tau$, $N$ and $m$ are the time step size, polynomial degree and regularity in the space variable of the exact solution, respectively. It should be noted that the global truncation error $\mathcal{O}(\tau^2+N^{-m})$ is well established in [ Li, Lv and Xu, {\em Numer. Methods Partial Differ. Equ}. (2019)]. Finally, some numerical experiments are carried out to verify the theoretical analysis. To the best of our knowledge, this is the first proof for the stability of the third-order scheme for the Caputo-Fabrizio fractional operator.