A numerical study for tempered time-fractional advection-dispersion equation on graded meshes

TL;DR

This paper introduces a second-order accurate, efficient numerical scheme for the tempered time-fractional advection-dispersion equation using a sum-of-exponentials approximation and graded meshes, reducing computational costs while maintaining accuracy.

Contribution

The paper develops a novel SOE-based time-stepping scheme with graded meshes for improved efficiency and accuracy in solving tempered time-fractional PDEs, with rigorous analysis and numerical validation.

Findings

Achieves same convergence order as classical schemes with less storage and computation.

Reduces storage from O(MN) to O(MN_exp) and computational complexity from O(MN^2) to O(MN N_exp).

Numerical results confirm the scheme's accuracy and efficiency.

Abstract

In this paper, we develop a second-order accurate time-stepping scheme for the tempered time-fractional advection-dispersion equation based on a sum-of-exponentials (SOE) approximation to the convolution kernel involved in the fractional derivative. To effectively resolve the weak initial-time singularity at t=0, graded temporal meshes are employed. A fully discrete scheme is constructed by coupling the proposed half-time-level temporal discretization with a finite difference method in space. Compared with the classical L1 scheme, the proposed SOE-based method achieves the same global convergence order while reducing both storage requirements and computational cost. Specifically, the storage demand is reduced from O(MN) to O(MN_exp), and the computational complexity is lowered from O(MN^2) to O(MN N_exp), where M and N denote the numbers of spatial and temporal grid points, respectively, and N_exp is the number of exponential terms used in the SOE approximation. The unique solvability, stability and accuracy of the resulting scheme are rigorously analyzed. Several numerical results are presented to confirm the sharpness of the error analysis and to demonstrate the efficiency of the proposed method.