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

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.

💡 Research Summary

This paper presents a second‑order accurate time‑stepping method for the tempered time‑fractional advection‑dispersion equation (TTFADE) by exploiting a sum‑of‑exponentials (SOE) approximation of the convolution kernel and graded temporal meshes. The governing equation combines a first‑order time derivative with a tempered Caputo fractional derivative of order α∈(0,1) and a spatial advection‑diffusion operator. The authors first introduce a graded mesh tₙ = T (n/N)ʳ (r≥1) to resolve the weak singularity at t=0, and they evaluate the fractional derivative at the half‑time level \bar tₙ = (tₙ + tₙ₊₁)/2. The kernel (t‑s)^{‑α}e^{‑λ(t‑s)} is approximated by a finite sum ∑_{ℓ=1}^{N_exp} ω_ℓ e^{‑s_ℓ(t‑s)} with a prescribed tolerance ε. This SOE representation yields a recursive formula for the history term, reducing storage from O(MN) to O(MN_exp) and computational cost from O(MN²) to O(MN N_exp), where M and N are the numbers of spatial and temporal grid points.

The history part is combined with a local part approximated by linear interpolation (L1) on each sub‑interval, leading to a compact expression for the discrete fractional derivative at \bar tₙ. Spatial discretization uses a standard second‑order central finite‑difference scheme with homogeneous Dirichlet boundary conditions, resulting in a fully discrete linear system at each time step.

Rigorous analysis is provided. Lemma 2.1 establishes the truncation error of the L1 approximation on graded meshes, showing it decays like N^{‑r(δ+1‑α)} for the first step and N^{‑min{2‑α, r(1+δ)}} thereafter, where δ characterizes the solution’s initial‑time regularity. Theorem 2.1 extends this to the SOE‑based operator, adding the SOE tolerance ε. Stability is proved via an energy method, demonstrating unconditional L² stability under the mesh grading condition r≥1. Uniqueness of the discrete solution follows from the positive definiteness of the resulting matrix.

Three numerical experiments validate the theory. Test 1 with a smooth exact solution confirms second‑order convergence in both time and space. Test 2, featuring a solution with a weak singularity (δ≈1.5), still achieves the predicted rates when r=2. Test 3 illustrates the efficiency gains: for M=N=2000 and N_exp≈8, memory usage drops to roughly one‑twelfth of the classical L1 scheme and CPU time is reduced to about 35 % of the original cost, while preserving accuracy.

In summary, the proposed SOE‑graded‑mesh method delivers a high‑order, memory‑efficient, and computationally cheap solver for TTFADEs, handling weak initial singularities without sacrificing stability. The authors suggest extensions to multi‑dimensional problems, non‑uniform spatial grids, and nonlinear reaction terms as future work.