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Higher-Order Finite Difference Methods for the Tempered Fractional Laplacian

Mingyi Wang, Dongling Wang

TL;DR

This work addresses the numerical solution of tempered fractional Laplacian equations on bounded domains by developing high-order finite difference schemes (HFD) that leverage generating functions from the classical Laplacian and a semi-discrete Fourier transform. The resulting discrete operator $(-\Delta_h)_{\lambda}^{\frac{\alpha}{2}}$ uses weights $a_{j}^{(\alpha,h\lambda)}$ to form a Toeplitz stiffness matrix, enabling FFT acceleration. The authors establish $l^{\infty}$-convergence of order $p\in\{4,6,8\}$ for smooth solutions and prove stability with $l^{2}$-convergence, supported by a detailed error analysis of the discrete symbol $S^{\alpha}_{\lambda,h}(\xi)$. Numerical experiments in multiple dimensions confirm the predicted rates and demonstrate the advantages of higher-order schemes over second-order baselines, with practical implications for efficiently solving nonlocal tempered diffusion problems.

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.

Higher-Order Finite Difference Methods for the Tempered Fractional Laplacian

TL;DR

This work addresses the numerical solution of tempered fractional Laplacian equations on bounded domains by developing high-order finite difference schemes (HFD) that leverage generating functions from the classical Laplacian and a semi-discrete Fourier transform. The resulting discrete operator uses weights to form a Toeplitz stiffness matrix, enabling FFT acceleration. The authors establish -convergence of order for smooth solutions and prove stability with -convergence, supported by a detailed error analysis of the discrete symbol . Numerical experiments in multiple dimensions confirm the predicted rates and demonstrate the advantages of higher-order schemes over second-order baselines, with practical implications for efficiently solving nonlocal tempered diffusion problems.

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 . 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.
Paper Structure (10 sections, 8 theorems, 74 equations, 7 figures, 11 tables, 2 algorithms)

This paper contains 10 sections, 8 theorems, 74 equations, 7 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The HFD schemes for the TFL
  3. The approximation for the TFL
  4. Convergence analysis for the discrete TFL
  5. The HFD methods for TFL equations
  6. Numerical accuracy
  7. The approximation accuracy for the coefficients $a_j^{(\alpha,h\lambda)}$
  8. The approximation accuracy for the TFL $(-\Delta)_{\lambda}^{\frac{\alpha}{2}}$
  9. The numerical approximations for the TFL equations
  10. Conclusions

Key Result

Lemma 2.1

(Symbol estimates) For the frequency-domain symbol $S^{\alpha}_{\lambda}(\xi)$ of the TFL in eq:1.5 and its discrete counterpart $S^{\alpha}_{\lambda,h}(\xi)$ in eq:2.8, there exists a positive constant $C$, independent of the spatial step size $h$, such that the following estimates hold where $p=4, 6, 8.$

Figures (7)

  • Figure 1: Numerical errors $e_{\infty}(h)$ and $e_{l^2}(h)$ of $(-\Delta_h)_{\lambda}^{\frac{\alpha}{2}}u(x_1,x_2)$ with $s=5+\alpha$ in \ref{['exa1']}.
  • Figure 2: Numerical errors $e_{\infty}(h)$ and $e_{l^2}(h)$ of $(-\Delta_h)_{\lambda}^{\frac{\alpha}{2}}u(x_1,x_2)$ with $s=2+\alpha$ in \ref{['exa1']}.
  • Figure 3: TFL $(-\Delta)_{\lambda}^{\frac{\alpha}{2}}u(x_1,x_2)$ in \ref{['exa1']} ($\alpha=0.4,\ s=2+\alpha, 5+\alpha$).
  • Figure 4: Numerical error $E_{\infty}(h)$ in \ref{['exa2']} ($\lambda= 0.2, 0.5, 1.0, 1.5, 3.0, 5.0,\ \alpha=0.4, 1.8).$
  • Figure 5: Numerical accuracy of nonmesh points in \ref{['exa2']} ($\lambda = 0.5,\ \alpha = 0.4,\ 1.8,\ s = 6$).
  • ...and 2 more figures

Theorems & Definitions (23)

  • Definition 2.1
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.2
  • Theorem 2.1
  • proof
  • Definition 3.1
  • Lemma 3.1
  • ...and 13 more