一种用于回火分数阶拉普拉斯算子的加权积分正则化有限差分格式
A Weighted Integral-Regularized Finite Difference Scheme for the Tempered Fractional Laplacian
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中文总结 AI 辅助
针对回火分数阶拉普拉斯算子数值方法开发难题，提出加权积分正则化有限差分法，通过泰勒展开正则化奇异积分，推导多维算子截断误差界，建立一维方程稳定性和收敛性，经数值实验验证方法的精确、高效与稳健。
中文摘要 AI 辅助
回火分数阶拉普拉斯算子（TFL）的内在奇异性给同时精确、高效且易于实现的数值方法开发带来重大挑战。我们提出加权积分正则化有限差分（WIRFD）方法，通过含光滑窗函数的多维泰勒展开正则化奇异被积函数。将所得积分分解为用穿孔梯形法则离散的正则项和直接计算修正项。对于多维TFL算子，引入光滑辅助函数和混叠公式得出\(O(h^{4 - \alpha})\)截断误差界。对于一维TFL方程，基于离散矩阵严格对角占优及其最小特征值下界建立稳定性和最优收敛性。离散矩阵的托普利兹结构使基于FFT矩阵向量乘法成为可能，用预处理共轭梯度法高效求解线性系统。数值实验证实理论结果，证明该方法精确、高效且稳健。
英文摘要
The intrinsic singularity of the tempered fractional Laplacian (TFL) remains a major challenge in developing numerical methods that are simultaneously accurate, efficient, and easy to implement. We develop a weighted integral-regularized finite difference (WIRFD) method that regularizes the singular integrand via a multidimensional Taylor expansion incorporating a smooth window function. The resulting integral is decomposed into a regularized term, which is discretized by a punctured trapezoidal rule, and a directly evaluated correction term. For the multidimensional TFL operator, we derive an $O(h^{4-α})$ truncation error bound in the $l^{\infty}$-norm for $α\in(0,2)$ and $u\in C^s(\mathbb{R}^d)$ with $s\geq 8$ by introducing a smooth auxiliary function together with the aliasing formula. For the one-dimensional TFL equation, we establish stability in both the $l^2$- and $l^{\infty}$-norms and optimal $O(h^{4-α})$ convergence for $α\in[1,2)$ based on the strict diagonal dominance of the discrete matrix and a lower bound for its minimum eigenvalue. The Toeplitz structure of the discrete matrix enables FFT-based matrix-vector multiplication, and the resulting linear systems are solved efficiently by a preconditioned conjugate gradient (PCG) method. Numerical experiments corroborate the theoretical results, demonstrating the accuracy, efficiency, and robustness of the proposed method.
