A New Fast Finite Difference Scheme for Tempered Time Fractional Advection-Dispersion Equation with a Weak Singularity at Initial Time
Liangcai Huang, Shujuan Lü
TL;DR
This work develops a fast, second-order Crank–Nicolson-type finite-difference scheme for the tempered time-fractional advection-dispersion equation, addressing weak initial-time singularities through a graded temporal mesh and a history/local decomposition. A novel fast evaluation operator for the Caputo tempered derivative is derived using exponential-sum approximations and linear interpolation, reducing computational cost. The authors provide rigorous error analysis and prove unique solvability, stability, and convergence of the fully discrete scheme, with numerical experiments confirming second-order accuracy in both time and space. The approach is especially applicable to problems with memory effects and nonsmooth initial data, such as groundwater transport modeling.
Abstract
In this paper, we propose a new second-order fast finite difference scheme in time for solving the Tempered Time Fractional Advection-Dispersion Equation. Under the assumption that the solution is nonsmooth at the initial time, we investigate the uniqueness, stability, and convergence of the scheme. Furthermore, we prove that the scheme achieves second-order convergence in both time and space. Finally, corresponding numerical examples are provided.