Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications

In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ ($0 < \alpha < 2$) in hypersingular integral form. The proposed finite difference methods provide a fractional analogue of the central difference schemes to the fractional Laplacian, and as $\alpha \to 2^-$, they collapse to the central difference schemes of the classical Laplace operator $-\Delta$. We prove that our methods are consistent if $u \in C^{\lfloor\alpha\rfloor, \alpha-\lfloor\alpha\rfloor+\epsilon}({\mathbb R}^d)$, and the local truncation error is ${\mathcal O}(h^\epsilon)$, with $\epsilon > 0$ a small constant and $\lfloor \cdot \rfloor$ denoting the floor function. If $u \in C^{2+\lfloor\alpha\rfloor, \alpha-\lfloor\alpha\rfloor+\epsilon}({\mathbb R}^d)$, they can achieve the second order of accuracy for any $\alpha \in (0, 2)$. These results hold for any dimension $d \ge 1$ and thus improve the existing error estimates for the finite difference method of the one-dimensional fractional Laplacian. 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 {\it at most} satisfy $u \in C^{1,1}({\mathbb R}^d)$. 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.Comment: 24 pages, 6 figures, and 6 table

Numerical investigation on nonlocal problems with the fractional Laplacian

Nonlocal models have recently become a powerful tool for studying complex systems with long-range interactions or memory effects, which cannot be described properly by the traditional differential equations. So far, different nonlocal (or fractional differential) models have been proposed, among which models with the fractional Laplacian have been well applied. The fractional Laplacian (-Δ)α/2 represents the infinitesimal generator of a symmetric α-stable Lévy process. It has been used to describe anomalous diffusion, turbulent flows, stochastic dynamics, finance, and many other phenomena. However, the nonlocality of the fractional Laplacian introduces considerable challenges in its mathematical modeling, numerical simulations, and mathematical analysis. To advance the understanding of the fractional Laplacian, two novel and accurate finite difference methods -- the weighted trapezoidal method and the weighted linear interpolation method are developed for discretizing the fractional Laplacian. Numerical analysis is provided for the error estimates, and fast algorithms are developed for their efficient implementation. Compared to the current state-of-the-art methods, these two methods have higher accuracy but less computational complexity. As an application, the solution behaviors of the fractional Schördinger equation are investigated to understand the nonlocal effects of the fractional Laplacian. First, the eigenvalues and eigenfunctions of the fractional Schrödinger equation in an infinite potential well are studied, and the results provide insights into an open problem in the fractional quantum mechanics. Second, three Fourier spectral methods are developed and compared in solving the fractional nonlinear Schördinger equation (NLS), among which the SSFS method is more effective in the study of the plane wave dynamics. Sufficient conditions are provided to avoid the numerical instability of the SSFS method. In contrast to the standard NLS, the plane wave dynamics of the fractional NLS are more chaotic due to the long-range interactions --Abstract, page iii

Numerical approximations for the tempered fractional Laplacian: Error analysis and applications

In this paper, we propose an accurate finite difference method to discretize the $d$-dimensional (for $d\ge 1$) 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 $O(h^\epsilon)$, for $u \in C^{0, \alpha+\epsilon} (\bar{\Omega})$ if $\alpha < 1$ (or $u \in C^{1, \alpha-1+\epsilon} (\bar{\Omega})$ if $\alpha \ge 1$) with $\epsilon > 0$, suggesting the minimum consistency conditions. The accuracy can be improved to $O(h^2)$, for $u \in C^{2, \alpha+\epsilon} (\bar{\Omega})$ if $\alpha < 1$ (or $u \in C^{3, \alpha - 1 + \epsilon} (\bar{\Omega})$ if $\alpha \ge 1$). 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 solution $u \in C^{1,1}(\bar{\Omega})$ for any $0<\alpha<2$. Moreover, if the solution of tempered fractional Poisson problems satisfies $u \in C^{p, s}(\bar{\Omega})$ for $p = 0, 1$ and $0<s \le 1$, our method has the accuracy of $O(h^{p+s})$. 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.Comment: 21 pages, 11 figures, 3 table

Computing the Ground and First Excited States of Fractional Schrodinger Equations in an Inifinite Potential Well

In this paper, we numerically study the ground and first excited states of the fractional Schrodinger equation in an infinite potential well. Due to the non-locality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrodinger equation either analytically or numerically. We first introduce a fractional gradient flow with discrete normalization and then discretize it by using the trapezoidal type quadrature rule in space and the semi-implicit Euler method in time. Our method can be used to compute the ground and first excited states not only in the linear cases but also in the nonlinear cases. Moreover, it can be generalized to solve the fractional partial differential equations (PDEs) with Riesz fractional derivatives in space. Our numerical results suggest that the eigenfunctions of the fractional Schrodinger equation in an infinite potential well are significantly different from those of the standard (non-fractional) Schrodinger equation. In addition, we find that the strong nonlocal interactions represented by the fractional Laplacian can lead to a large scattering of particles inside of the potential well. Compared to the ground states, the scattering of particles in the first excited states is larger. Furthermore, boundary layers emerge in the ground states and additionally inner layers exist in the first excited states of the fractional nonlinear Schrodinger equation

Mass Conservative Method for Solving the Fractional Nonlinear Schrödinger Equation

We propose three Fourier spectral methods, i.e., the split-step Fourier spectral (SSFS), the Crank–Nicolson Fourier spectral (CNFS), and the relaxation Fourier spectral (ReFS) methods, for solving the fractional nonlinear Schrödinger (NLS) equation. All of them are mass conservative and time reversible, and they have the spectral order accuracy in space and the second-order accuracy in time. In addition, the CNFS and ReFS methods are energy conservative. The performance of these methods in simulating the plane wave and soliton dynamics is discussed. The SSFS method preserves the dispersion relation, and thus it is more accurate for studying the long-time behaviors of the plane wave solutions. Furthermore, our numerical simulations suggest that the SSFS method is better in solving the defocusing NLS, but the CNFS and ReFS methods are more effective for the focusing NLS

A Fast Algorithm for Solving the Space-Time Fractional Diffusion Equation

In this paper, we propose a fast algorithm for efficient and accurate solution of the space-time fractional diffusion equations defined in a rectangular domain. The spatial discretization is done by using the central finite difference scheme and matrix transfer technique. Due to its nonlocality, numerical discretization of the spectral fractional Laplacian (−Δ)sα/2 results in a large dense matrix. This causes considerable challenges not only for storing the matrix but also for computing matrix-vector products in practice. By utilizing the compact structure of the discrete system and the discrete sine transform, our algorithm avoids to store the large matrix from discretizing the nonlocal operator and also significantly reduces the computational costs. We then use the Laplace transform method for time integration of the semi-discretized system and a weighted trapezoidal method to numerically compute the convolutions needed in the resulting scheme. Various experiments are presented to demonstrate the efficiency and accuracy of our method

A Comparative Study on Nonlocal Diffusion Operators Related to the Fractional Laplacian

In this paper, we study four nonlocal diffusion operators, including the fractional Laplacian, spectral fractional Laplacian, regional fractional Laplacian, and peridynamic operator. These operators represent the infinitesimal generators of different stochastic processes, and especially their differences on a bounded domain are significant. We provide extensive numerical experiments to understand and compare their differences. We find that these four operators collapse to the classical Laplace operator as α 2. The eigenvalues and eigenfunctions of these four operators are different, and the k-th (for k σ N) eigenvalue of the spectral fractional Laplacian is always larger than those of the fractional Laplacian and regional fractional Laplacian. For any ff 2 (0; 2), the peridynamic operator can provide a good approximation to the fractional Laplacian, if the horizon size δ is suffciently large. We find that the solution of the peridynamic model converges to that of the fractional Laplacian model at a rate of O(δ-a). In contrast, although the regional fractional Laplacian can be used to approximate the fractional Laplacian as α 2, it generally provides inconsistent result from that of the fractional Laplacian if α ≤ 2. Moreover, some conjectures are made from our numerical results, which could contribute to the mathematics analysis on these operators

Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion

We analytically and numerically investigate the stability and dynamics of the plane wave solutions of the fractional nonlinear Schrödinger (NLS) equation, where the long-range dispersion is described by the fractional Laplacian (−∆)α/2 . The linear stability analysis shows that plane wave solutions in the defocusing NLS are always stable if the power α ∈ [1, 2] but unstable for α ∈ (0, 1). In the focusing case, they can be linearly unstable for any α ∈ (0, 2]. We then apply the split-step Fourier spectral (SSFS) method to simulate the nonlinear stage of the plane waves dynamics. In agreement with earlier studies of solitary wave solutions of the fractional focusing NLS, we find that as α ∈ (1, 2] decreases, the solution evolves towards an increasingly localized pulse existing on the background of a sea of small-amplitude dispersive waves. Such a highly localized pulse has a broad spectrum, most of whose modes are excited in the nonlinear stage of the pulse evolution and are not predicted by the linear stability analysis. For α ≤ 1, we always find the solution to undergo collapse. We also show, for the first time to our knowledge, that for initial conditions with nonzero group velocities (traveling plane waves), an onset of collapse is delayed compared to that for a standing plane wave initial condition. For defocusing fractional NLS, even though we find traveling plane waves to be linearly unstable for α \u3c 1, we have never observed collapse. As a by-product of our numerical studies, we derive a stability condition on the time step of the SSFS to guarantee that this method is free from numerical instabilities

A Novel and Accurate Finite Difference Method for the Fractional Laplacian and the Fractional Poisson Problem

In this paper, we develop a novel finite difference method to discretize the fractional Laplacian (−Δ)α/2 in hypersingular integral form. By introducing a splitting parameter, we formulate the fractional Laplacian as the weighted integral of a weak singular function, which is then approximated by the weighted trapezoidal rule. Compared to other existing methods, our method is more accurate and simpler to implement, and moreover it closely resembles the central difference scheme for the classical Laplace operator. We prove that for u ∈ C3,α/2(R), our method has an accuracy of O(h2) uniformly for any α ∈ (0,2), while for u ∈ C1,α/2(R), the accuracy is O(1-α/2). The convergence behavior of our method is consistent with that of the central difference approximation of the classical Laplace operator. Additionally, we apply our method to solve the fractional Poisson equation and study the convergence of its numerical solutions. The extensive numerical examples that accompany our analysis verify our results, as well as give additional insights into the convergence behavior of our method