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Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains
SIAM Journal on Numerical Analysis ( IF 2.9 ) Pub Date : 2020-01-01 , DOI: 10.1137/19m128377x
Changtao Sheng , Jie Shen , Tao Tang , Li-Lian Wang , Huifang Yuan

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalised, and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., $O((N\log_2N)^d)$ with $N$ being the number of modes in each spatial direction. Ample numerical tests for various decaying exact solutions show that the convergence of the fast solver perfectly matches the order of theoretical error estimates. With a suitable time-discretization, the fast solver can be directly applied to a large class of nonlinear fractional PDEs. As an example, we solve the fractional nonlinear Schr{o}dinger equation by using the fourth-order time-splitting method together with the proposed MCF-spectral-Galerkin method.

中文翻译:

用于无界域中具有积分分数拉普拉斯算子的偏微分方程的快速傅立叶类映射切比雪夫谱-伽辽金方法

在本文中,我们提出了一种快速谱伽辽金方法,用于求解 $\mathbb{R}^d$ 中涉及积分分数拉普拉斯算子的偏微分方程,该方法建立在两个基本组成部分上:(i) 分数拉普拉斯算子的 Dunford-Taylor 公式; (ii) 类傅立叶双正交映射切比雪夫函数 (MCF) 作为基函数。因此,分数拉普拉斯算子可以完全对角化,求解椭圆分数 PDE 的复杂度是拟最优的,即 $O((N\log_2N)^d)$ 其中 $N$ 是每个空间方向。对各种衰减精确解的大量数值测试表明,快速求解器的收敛性与理论误差估计的阶数完全匹配。通过合适的时间离散化,快速求解器可以直接应用于一大类非线性分数阶偏微分方程。
更新日期:2020-01-01
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