Higher-Order Finite Difference Methods for the Tempered Fractional Laplacian

TL;DR

This paper develops high-order finite difference schemes for the tempered fractional Laplacian, achieving high accuracy and efficiency, with rigorous analysis and numerical validation confirming their effectiveness.

Contribution

It introduces a general framework for high-order finite difference methods for TFL using new generating functions, enabling efficient and accurate solutions.

Findings

Achieves high-order convergence rates of 4, 6, and 8.

Produces Toeplitz matrices enabling fast computations.

Numerical results confirm theoretical accuracy and stability.

Abstract

This paper presents a general framework of high-order finite difference (HFD) schemes for the tempered fractional Laplacian (TFL) based on new generating functions obtained from the discrete symbols. Specifically, for sufficiently smooth functions, the resulting discretizations achieve high-order convergence with orders $p=4, 6, 8$. The discrete operators lead to Toeplitz stiffness matrices, allowing efficient matrix-vector multiplications via fast algorithms. Building on these approximations, HFD methods are formulated for solving TFL equations, and their stability and convergence are rigorously analyzed. Numerical simulations confirm the effectiveness of the proposed methods, showing excellent agreement with the theoretical predictions.