Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives
Bin Fan
TL;DR
The paper addresses efficiently solving multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives by marrying a fast, memory-saving L1-type time discretization with a shifted Legendre spectral collocation for space. The resulting fully discrete scheme is unconditionally stable and achieves temporal order $2$ and spectral spatial convergence, with storage $O(1)$ and temporal complexity $O(N_T)$. Rigorous stability and error analyses are provided for the semi- and full-discretizations, backed by numerical experiments confirming accuracy and efficiency. The approach scales well to higher dimensions and offers a practical pathway for high-resolution simulations of fractional diffusion processes.
Abstract
In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders $α_i\in(0,1)$, $i=1,2,\cdots,n$). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only $O(1)$ storage and $O(N_T)$ computational complexity, where $N_T$ denotes the total number of time steps. Then we use a Legendre spectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established. We demonstrate that the proposed scheme is unconditionally stable and convergent with an order of $O(\left(Δt\right)^{2}+N^{-m})$, where $Δt$, $N$, and $m$ represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively. Numerical results are presented to validate the theoretical predictions.