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

Liangcai Huang, Lin Li, Shujuan Lü

TL;DR

This paper addresses the numerical solution of the tempered time-fractional advection-dispersion equation with weak initial-time singularities by developing a second-order time-stepping method based on a sum-of-exponentials (SOE) approximation of the tempered Caputo kernel on graded temporal meshes. It couples this with a finite-difference spatial discretization in a Crank-Nicolson-type framework and uses half-time-level evaluation to achieve second-order temporal accuracy. The proposed approach dramatically reduces storage from ${\mathcal O}(MN)$ to ${\mathcal O}(MN_{exp})$ and computational cost from ${\mathcal O}(MN^2)$ to ${\mathcal O}(M N N_{exp})$, with ${N_{exp}}=\mathcal{O}(\log N)$, while preserving the same global convergence order as standard L1 schemes. Rigorous solvability, stability, and convergence results are derived, and numerical experiments across multiple cases confirm sharp error estimates and substantial efficiency gains, particularly for long-time simulations.

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.

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

TL;DR

This paper addresses the numerical solution of the tempered time-fractional advection-dispersion equation with weak initial-time singularities by developing a second-order time-stepping method based on a sum-of-exponentials (SOE) approximation of the tempered Caputo kernel on graded temporal meshes. It couples this with a finite-difference spatial discretization in a Crank-Nicolson-type framework and uses half-time-level evaluation to achieve second-order temporal accuracy. The proposed approach dramatically reduces storage from to and computational cost from to , with , while preserving the same global convergence order as standard L1 schemes. Rigorous solvability, stability, and convergence results are derived, and numerical experiments across multiple cases confirm sharp error estimates and substantial efficiency gains, particularly for long-time simulations.

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.
Paper Structure (8 sections, 100 equations, 3 figures, 7 tables)

This paper contains 8 sections, 100 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The discrete scheme and error estimation to the Tempered Caputo fractional Derivative
  3. Construction of the fully discrete scheme and the well-posedness
  4. Construction of the fully discrete scheme
  5. The unique solvability and stability of the fully discrete scheme
  6. Convergence of the fully discrete scheme
  7. Numerical Experiments
  8. Concluding remarks

Figures (3)

  • Figure 4.1: (Left) A comparison of CPU time in seconds as a function of time steps $N$. (Right) SOE number $N_{exp}$ versus the time steps $N$.
  • Figure 4.2: A comparison of CPU time in seconds as a function of time steps $N$.
  • Figure 4.3: A comparison of CPU time in seconds as a function of time steps $N$.

Theorems & Definitions (6)

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