Higher-Order Finite Difference Methods for the Tempered Fractional Laplacian

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.

💡 Research Summary

This paper introduces a systematic framework for constructing high‑order finite‑difference (HFD) schemes for the tempered fractional Laplacian (TFL) operator ((-\Delta)^{\alpha}{\lambda}), where (\alpha\in(0,2)\setminus{1}) and (\lambda\ge0). The authors start from classical high‑order central difference formulas for the ordinary Laplacian (orders (p=4,6,8)) and derive their generating functions. By applying a semi‑discrete Fourier transform to these generating functions, they obtain discrete symbols that approximate the continuous frequency‑domain symbol (S{\alpha,\lambda}(\xi)) of the TFL.

The discrete TFL operator is expressed as
\