Numerical Approximations for the Tempered Fractional Laplacian: Error Analysis and Applications
Numerical Approximations for the Tempered Fractional Laplacian: Error Analysis and Applications
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DOI：
10.1007/s10915-019-01029-7
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发表时间：
2018-08
影响因子：
2.5
通讯作者：
Siwei Duo;Yanzhi Zhang
中科院分区：
文献类型：
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作者：
Siwei Duo;Yanzhi Zhang
在本文中，我们提出了一种精确的有限差分方法来离散分数次拉普拉斯积分，并用它来研究回火对各种应用中出现的问题求解的影响。与已有方法相比，该方法具有更高的精度和更简单的实现。我们的数值方法具有FORIF(ORIF)WITH的精度，建议了最小相容条件。精度可提高到，FORIF(ORIF)。数值实验证实了我们的分析结果，并为解决回火分数泊松问题提供了启示。它表明，为了达到二阶精度，我们的方法只需要任何解。此外，如果回火分数次Poisson问题的解满足，则我们的方法具有如下精度。由于我们的方法产生一个(多层)Toeplitz刚度矩阵，所以可以通过快速傅立叶变换来设计快速算法来进行有效的模拟。最后，我们将它与快速算法结合起来，研究了回火对各种回火分数阶偏微分方程解的影响，包括Allen-Cahn方程和Gray-Scott方程。
In this paper, we propose an accurate finite difference method to discretize thed-dimensional (for) tempered integral fractional Laplacian and apply it to study the tempered effects on the solution of problems arising in various applications. Compared to other existing methods, our method has higher accuracy and simpler implementation. Our numerical method has an accuracy of, forif(orif) with, suggesting the minimum consistency conditions. The accuracy can be improved to, forif(orif). Numerical experiments confirm our analytical results and provide insights in solving the tempered fractional Poisson problem. It suggests that to achieve the second order of accuracy, our method only requires the solutionfor any. Moreover, if the solution of tempered fractional Poisson problems satisfiesforand, our method has the accuracy of. Since our method yields a (multilevel) Toeplitz stiffness matrix, one can design fast algorithms via the fast Fourier transform for efficient simulations. Finally, we apply it together with fast algorithms to study the tempered effects on the solutions of various tempered fractional PDEs, including the Allen–Cahn equation and Gray–Scott equations.