A novel fast tempered algorithm with high-accuracy scheme for 2D tempered fractional reaction-advection-subdiffusion equation

Abstract

In this article, we propose a novel and computationally efficient difference scheme for evaluating the Caputo tempered fractional derivative (TFD). Our approach introduces a new fast tempered FλL2−1σ difference method, achieving a higher convergence rate of order O(Δt3−α). Specifically, we apply this method to a class of two-dimensional tempered-fractional reaction-advection-subdiffusion equations (2D-TF-RASDE) characterized by the tempered parameter λ and a tempered fractional derivative of order α, (where 0<α<1). The novel fast tempered scheme is based on the sum of exponents (SOE) technique. Additionally, we develop a novel high order compact finite difference (CFD) scheme using an alternating direction implicit (ADI) formulation for the 2D-TF-RASDE. We investigate the stability and convergence of this FLλ2−1σ-ADI-CD scheme. Numerical simulations reveal a convergence order of O(Δt1+α+hϰ4+hy4+ϵ) under robust regularity assumptions underlining the scheme's superior computational efficiency. Furthermore, our computational results align well with theoretical analysis, demonstrating both accuracy and reduced computational complexity and storage requirements as compared to the standard tempered Lλ2−1σ-ADI-CD scheme. Notably, the fast tempered FLλ2−1σ-ADI-CD scheme exhibits competitive performance and reduced CPU time relative to the standard Lλ2−1σ-ADI-CD scheme, thereby offering a distinct advantage for efficiently solving complex fractional differential equations.