Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications
Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications
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DOI：
10.1016/j.cma.2019.06.016
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发表时间：
2018-04
影响因子：
7.2
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，我们提出了准确且有效的有限差分方法，以超奇异积分形式离散化二维和三维分数拉普拉斯 (− Δ) α 2 (0< α< 2)。所提出的方法为分数拉普拉斯提供了中心差分格式的分数模拟。当 α → 2− 时，它们崩溃为经典拉普拉斯算子− Δ 的中心差分格式。我们证明，如果 u∈ C⌊ α⌋, α−⌊ α⌋+ ε (R d)，并且局部截断误差为 O (h ε)，且 ε> 0 是一个小常数，⌊⋅⌋ 表示下限函数，则我们的方法是一致的。如果 u∈ C 2+⌊ α⌋, α−⌊ α⌋+ ε (R d)，则对于任意 αε(0, 2)，它们可以达到二阶精度。这些结果对于任何维度 d≥ 1 都成立，从而改进了文献中一维情况的现有误差估计。提供了大量的数值实验并证实了我们的分析结果。然后，我们应用我们的方法来求解分数泊松问题和分数艾伦-卡恩方程。数值模拟表明，要达到二阶精度，分数阶泊松问题的解至多应满足 u∈ C 1, 1 (R d)。我们的方法的一个优点是它们产生了多级托普利茨刚度矩阵，这是通过快速傅里叶变换（FFT）开发快速算法的一个有吸引力的特性。我们对二维和三维分数艾伦-卡恩方程的研究证明了我们的方法在解决高维分数问题方面的效率。
In this paper, we propose accurate and efficient finite difference methods to discretize the two-and three-dimensional fractional Laplacian (− Δ) α 2 (0< α< 2) in hypersingular integral form. The proposed methods provide a fractional analogue of the central difference schemes to the fractional Laplacian. As α→ 2−, they collapse to the central difference schemes of the classical Laplace operator− Δ. We prove that our methods are consistent if u∈ C⌊ α⌋, α−⌊ α⌋+ ε (R d), and the local truncation error is O (h ε), with ε> 0 a small constant and⌊⋅⌋ denoting the floor function. If u∈ C 2+⌊ α⌋, α−⌊ α⌋+ ε (R d), they can achieve the second order of accuracy for any α∈(0, 2). These results hold for any dimension d≥ 1 and thus improve the existing error estimates of the one-dimensional cases in the literature. Extensive numerical experiments are provided and confirm our analytical results. We then apply our method to solve the fractional Poisson problems and the fractional Allen–Cahn equations. Numerical simulations suggest that to achieve the second order of accuracy, the solution of the fractional Poisson problem should at most satisfy u∈ C 1, 1 (R d). One merit of our methods is that they yield a multilevel Toeplitz stiffness matrix, an appealing property for the development of fast algorithms via the fast Fourier transform (FFT). Our studies of the two-and three-dimensional fractional Allen–Cahn equations demonstrate the efficiency of our methods in solving the high-dimensional fractional problems.