Computers & mathematics with applications, Mar 1, 2018
In this paper, we propose a fast algorithm for efficient and accurate solution of the space-time ... more 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 (−∆) α/2 s 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.
In this paper, we propose an accurate finite difference method to discretize the d-dimensional (f... more In this paper, we propose an accurate finite difference method to discretize the d-dimensional (for d ≥ 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 ε), for u ∈ C 0, α+ε (Ω) if α < 1 (or u ∈ C 1, α−1+ε (Ω) if α ≥ 1) with ε > 0, suggesting the minimum consistency conditions. The accuracy can be improved to O(h 2), for u ∈ C 2, α+ε (Ω) if α < 1 (or u ∈ C 3, α−1+ε (Ω) if α ≥ 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 ∈ C 1,1 (Ω) for any α ∈ (0, 2). Moreover, if the solution of tempered fractional Poisson problems satisfies u ∈ C p,s (Ω) for p = 0, 1 and s ∈ (0, 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.
Communications in Computational Physics, Jul 30, 2015
In this paper, we numerically study the ground and first excited states of the fractional Schrödi... more In this paper, we numerically study the ground and first excited states of the fractional Schrödinger 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 Schrödinger 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 Schrödinger equation in an infinite potential well are significantly different from those of the standard (non-fractional) Schrödinger 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 Schrödinger equation.
In this paper, we develop a novel finite difference method to discretize the fractional Laplacian... more In this paper, we develop a novel finite difference method to discretize the fractional Laplacian (−) 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 ∈ C 3,α/2 (R), our method has an accuracy of O(h 2) uniformly for any α ∈ (0, 2), while for u ∈ C 1,α/2 (R), the accuracy is O(h 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.
We study a space-fractional Allen-Cahn phase-field model in which the fractional Laplacian is def... more We study a space-fractional Allen-Cahn phase-field model in which the fractional Laplacian is defined to be an infinitesimal generator of an α-stable Lévy process with 0 < α < 2. Compared with the Allen-Cahn model using the Riemann-Liouville (R-L) fractional derivatives in the literature, the current model enjoys the following advantages: (1) it can control the sharpness and the decaying behavior of the interface via the tunable order parameter α as the Riemann-Liouville fractional model does; (2) it generates numerical solutions with sharper interfaces than the Riemann-Liouville fractional Allen-Cahn model with the same 1 < α < 2, and it can generate numerical solutions with even sharper interfaces as it also applies to the case for 0 < α < 1; (3) as α → 2 − , it provides good approximations to the classical Allen-Cahn model as the fractional Riemann-Liouville model does; (4) it is rotational invariant; and (5) it has a much slower mass decaying rate than the Riemann-Liouville fractional model. We also propose and analyze a first order in time and second order in space energy stable finite difference scheme.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Computers & mathematics with applications, Jun 1, 2016
We propose three Fourier spectral methods, i.e., the split-step Fourier spectral (SSFS), the Cran... more 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 secondorder 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.
In this paper, we propose an accurate finite difference method to discretize the d-dimensional (f... more In this paper, we propose an accurate finite difference method to discretize the d-dimensional (for d ≥ 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 ε), for u ∈ C 0, α+ε (Ω) if α < 1 (or u ∈ C 1, α−1+ε (Ω) if α ≥ 1) with ε > 0, suggesting the minimum consistency conditions. The accuracy can be improved to O(h 2), for u ∈ C 2, α+ε (Ω) if α < 1 (or u ∈ C 3, α−1+ε (Ω) if α ≥ 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 ∈ C 1,1 (Ω) for any α ∈ (0, 2). Moreover, if the solution of tempered fractional Poisson problems satisfies u ∈ C p,s (Ω) for p = 0, 1 and s ∈ (0, 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.
In this paper, we propose accurate and efficient finite difference methods to discretize the twoa... more In this paper, we propose accurate and efficient finite difference methods to discretize the twoand three-dimensional fractional Laplacian (−∆) α 2 (0 < α < 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 α → 2 − , they collapse to the central difference schemes of the classical Laplace operator −∆. 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 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 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.
We analytically and numerically investigate the stability and dynamics of the plane wave solution... more 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 (−Δ)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 th...
In this paper, we propose an accurate finite difference method to discretize the two and three di... more In this paper, we propose an accurate finite difference method to discretize the two and three dimensional fractional Laplacian (-Δ)^α/2 in the hypersingular integral form and apply it to solve the fractional reaction-diffusion equations. The key idea of our method is to split the strong singular kernel function of the fractional Laplacian. Hence, we first formulate the fractional Laplacian as the weighted integral of a central difference quotient and then approximate it by the weighted trapezoidal rule. It is proved that for u ∈ C^3, α/2(R^d), our method has an accuracy of O(h^2), uniformly for any α∈ (0, 2), while for u ∈ C^1, α/2(R^d), the accuracy is O(h^1-α/2). As α→ 2^-, the convergence behavior of our method is consistent with that of the central difference approximation of the classical Laplace operator. This study would fill the gap in the literature on numerical methods for the high dimensional factional Laplacian. In addition, we apply our method to solve the fractional r...
In this paper, we propose new and accurate finite difference methods to discretize the d-dimensio... more In this paper, we propose new and accurate finite difference methods to discretize the d-dimensional (d >=1) tempered fractional Laplacian and apply them to study the tempered effects on the solution of the fractional problems. Our finite difference methods have an accuracy of O(h^ϵ), for u ∈ C^0,α+ϵ (R^d) if α < 1 (or u ∈ C^1,α-1+ϵ(R^d) if α >=1) with ϵ > 0, suggesting the minimum consistency conditions. This accuracy can be improved to O(h^2), for u ∈ C^2,α+ϵ(R^d) if α < 1 (or u ∈ C^3,α- 1+ϵ(R^d) if α >= 1). Numerical results confirm our analytical conclusions and provide further insights of our methods in solving the tempered fractional Poisson problem. It suggests that to achieve the second order of accuracy, our methods only require the solution u ∈ C^1,1(R^d) for any 0 < α < 2. Moreover, if the solution u ∈ C^p, s(R^d) for 0 <= p, s <=1, our methods have the accuracy of O(h^p+s) in solving the tempered Poisson problem. Compared to other existing m...
Discrete & Continuous Dynamical Systems - B, 2019
In this paper, we study four nonlocal diffusion operators, including the fractional Laplacian, sp... more 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 ∈ N) eigenvalue of the spectral fractional Laplacian is always larger than those of the fractional Laplacian and regional fractional Laplacian. For any α ∈ (0, 2), the peridynamic operator can provide a good approximation to the fractional Laplacian, if the horizon size δ is sufficiently large. We find that the solution of the peridynamic model converges to that of the fractional Laplacian model at a rate of O(δ −α). 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.
In this paper, we propose an accurate finite difference method to discretize the d-dimensional (f... more In this paper, we propose an accurate finite difference method to discretize the d-dimensional (for d ≥ 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 ε), for u ∈ C 0, α+ε (Ω) if α < 1 (or u ∈ C 1, α−1+ε (Ω) if α ≥ 1) with ε > 0, suggesting the minimum consistency conditions. The accuracy can be improved to O(h 2), for u ∈ C 2, α+ε (Ω) if α < 1 (or u ∈ C 3, α−1+ε (Ω) if α ≥ 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 ∈ C 1,1 (Ω) for any α ∈ (0, 2). Moreover, if the solution of tempered fractional Poisson problems satisfies u ∈ C p,s (Ω) for p = 0, 1 and s ∈ (0, 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.
Computer Methods in Applied Mechanics and Engineering, 2019
In this paper, we propose accurate and efficient finite difference methods to discretize the twoa... more In this paper, we propose accurate and efficient finite difference methods to discretize the twoand three-dimensional fractional Laplacian (−∆) α 2 (0 < α < 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 α → 2 − , they collapse to the central difference schemes of the classical Laplace operator −∆. 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 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 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.
We study a space-fractional Allen-Cahn phase-field model in which the fractional Laplacian is def... more We study a space-fractional Allen-Cahn phase-field model in which the fractional Laplacian is defined to be an infinitesimal generator of an α-stable Lévy process with 0 < α < 2. Compared with the Allen-Cahn model using the Riemann-Liouville (R-L) fractional derivatives in the literature, the current model enjoys the following advantages: (1) it can control the sharpness and the decaying behavior of the interface via the tunable order parameter α as the Riemann-Liouville fractional model does; (2) it generates numerical solutions with sharper interfaces than the Riemann-Liouville fractional Allen-Cahn model with the same 1 < α < 2, and it can generate numerical solutions with even sharper interfaces as it also applies to the case for 0 < α < 1; (3) as α → 2 − , it provides good approximations to the classical Allen-Cahn model as the fractional Riemann-Liouville model does; (4) it is rotational invariant; and (5) it has a much slower mass decaying rate than the Riemann-Liouville fractional model. We also propose and analyze a first order in time and second order in space energy stable finite difference scheme.
In this paper, we develop a novel finite difference method to discretize the fractional Laplacian... more In this paper, we develop a novel finite difference method to discretize the fractional Laplacian (−) 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 ∈ C 3,α/2 (R), our method has an accuracy of O(h 2) uniformly for any α ∈ (0, 2), while for u ∈ C 1,α/2 (R), the accuracy is O(h 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.
Computers & Mathematics with Applications, 2017
In this paper, we propose a fast algorithm for efficient and accurate solution of the space-time ... more 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 (−∆) α/2 s 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.
Computers & Mathematics with Applications, 2016
We propose three Fourier spectral methods, i.e., the split-step Fourier spectral (SSFS), the Cran... more 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 secondorder 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.
In this paper, we numerically study the ground and first excited states of the fractional Schrödi... more In this paper, we numerically study the ground and first excited states of the fractional Schrödinger equation in an infinite potential well. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrödinger equation analytically. We first introduce a normalized fractional gradient flow and then discretize it by a quadrature rule method in space and the semi-implicit Euler method in time. Our numerical results suggest that the eigenfunctions of the fractional Schrödinger equation in an infinite potential well differ from those of the standard (non-fractional) Schrödinger equation. 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 ...
Computers & mathematics with applications, Mar 1, 2018
In this paper, we propose a fast algorithm for efficient and accurate solution of the space-time ... more 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 (−∆) α/2 s 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.
In this paper, we propose an accurate finite difference method to discretize the d-dimensional (f... more In this paper, we propose an accurate finite difference method to discretize the d-dimensional (for d ≥ 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 ε), for u ∈ C 0, α+ε (Ω) if α < 1 (or u ∈ C 1, α−1+ε (Ω) if α ≥ 1) with ε > 0, suggesting the minimum consistency conditions. The accuracy can be improved to O(h 2), for u ∈ C 2, α+ε (Ω) if α < 1 (or u ∈ C 3, α−1+ε (Ω) if α ≥ 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 ∈ C 1,1 (Ω) for any α ∈ (0, 2). Moreover, if the solution of tempered fractional Poisson problems satisfies u ∈ C p,s (Ω) for p = 0, 1 and s ∈ (0, 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.
Communications in Computational Physics, Jul 30, 2015
In this paper, we numerically study the ground and first excited states of the fractional Schrödi... more In this paper, we numerically study the ground and first excited states of the fractional Schrödinger 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 Schrödinger 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 Schrödinger equation in an infinite potential well are significantly different from those of the standard (non-fractional) Schrödinger 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 Schrödinger equation.
In this paper, we develop a novel finite difference method to discretize the fractional Laplacian... more In this paper, we develop a novel finite difference method to discretize the fractional Laplacian (−) 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 ∈ C 3,α/2 (R), our method has an accuracy of O(h 2) uniformly for any α ∈ (0, 2), while for u ∈ C 1,α/2 (R), the accuracy is O(h 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.
We study a space-fractional Allen-Cahn phase-field model in which the fractional Laplacian is def... more We study a space-fractional Allen-Cahn phase-field model in which the fractional Laplacian is defined to be an infinitesimal generator of an α-stable Lévy process with 0 < α < 2. Compared with the Allen-Cahn model using the Riemann-Liouville (R-L) fractional derivatives in the literature, the current model enjoys the following advantages: (1) it can control the sharpness and the decaying behavior of the interface via the tunable order parameter α as the Riemann-Liouville fractional model does; (2) it generates numerical solutions with sharper interfaces than the Riemann-Liouville fractional Allen-Cahn model with the same 1 < α < 2, and it can generate numerical solutions with even sharper interfaces as it also applies to the case for 0 < α < 1; (3) as α → 2 − , it provides good approximations to the classical Allen-Cahn model as the fractional Riemann-Liouville model does; (4) it is rotational invariant; and (5) it has a much slower mass decaying rate than the Riemann-Liouville fractional model. We also propose and analyze a first order in time and second order in space energy stable finite difference scheme.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Computers & mathematics with applications, Jun 1, 2016
We propose three Fourier spectral methods, i.e., the split-step Fourier spectral (SSFS), the Cran... more 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 secondorder 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.
In this paper, we propose an accurate finite difference method to discretize the d-dimensional (f... more In this paper, we propose an accurate finite difference method to discretize the d-dimensional (for d ≥ 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 ε), for u ∈ C 0, α+ε (Ω) if α < 1 (or u ∈ C 1, α−1+ε (Ω) if α ≥ 1) with ε > 0, suggesting the minimum consistency conditions. The accuracy can be improved to O(h 2), for u ∈ C 2, α+ε (Ω) if α < 1 (or u ∈ C 3, α−1+ε (Ω) if α ≥ 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 ∈ C 1,1 (Ω) for any α ∈ (0, 2). Moreover, if the solution of tempered fractional Poisson problems satisfies u ∈ C p,s (Ω) for p = 0, 1 and s ∈ (0, 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.
In this paper, we propose accurate and efficient finite difference methods to discretize the twoa... more In this paper, we propose accurate and efficient finite difference methods to discretize the twoand three-dimensional fractional Laplacian (−∆) α 2 (0 < α < 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 α → 2 − , they collapse to the central difference schemes of the classical Laplace operator −∆. 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 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 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.
We analytically and numerically investigate the stability and dynamics of the plane wave solution... more 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 (−Δ)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 th...
In this paper, we propose an accurate finite difference method to discretize the two and three di... more In this paper, we propose an accurate finite difference method to discretize the two and three dimensional fractional Laplacian (-Δ)^α/2 in the hypersingular integral form and apply it to solve the fractional reaction-diffusion equations. The key idea of our method is to split the strong singular kernel function of the fractional Laplacian. Hence, we first formulate the fractional Laplacian as the weighted integral of a central difference quotient and then approximate it by the weighted trapezoidal rule. It is proved that for u ∈ C^3, α/2(R^d), our method has an accuracy of O(h^2), uniformly for any α∈ (0, 2), while for u ∈ C^1, α/2(R^d), the accuracy is O(h^1-α/2). As α→ 2^-, the convergence behavior of our method is consistent with that of the central difference approximation of the classical Laplace operator. This study would fill the gap in the literature on numerical methods for the high dimensional factional Laplacian. In addition, we apply our method to solve the fractional r...
In this paper, we propose new and accurate finite difference methods to discretize the d-dimensio... more In this paper, we propose new and accurate finite difference methods to discretize the d-dimensional (d >=1) tempered fractional Laplacian and apply them to study the tempered effects on the solution of the fractional problems. Our finite difference methods have an accuracy of O(h^ϵ), for u ∈ C^0,α+ϵ (R^d) if α < 1 (or u ∈ C^1,α-1+ϵ(R^d) if α >=1) with ϵ > 0, suggesting the minimum consistency conditions. This accuracy can be improved to O(h^2), for u ∈ C^2,α+ϵ(R^d) if α < 1 (or u ∈ C^3,α- 1+ϵ(R^d) if α >= 1). Numerical results confirm our analytical conclusions and provide further insights of our methods in solving the tempered fractional Poisson problem. It suggests that to achieve the second order of accuracy, our methods only require the solution u ∈ C^1,1(R^d) for any 0 < α < 2. Moreover, if the solution u ∈ C^p, s(R^d) for 0 <= p, s <=1, our methods have the accuracy of O(h^p+s) in solving the tempered Poisson problem. Compared to other existing m...
Discrete & Continuous Dynamical Systems - B, 2019
In this paper, we study four nonlocal diffusion operators, including the fractional Laplacian, sp... more 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 ∈ N) eigenvalue of the spectral fractional Laplacian is always larger than those of the fractional Laplacian and regional fractional Laplacian. For any α ∈ (0, 2), the peridynamic operator can provide a good approximation to the fractional Laplacian, if the horizon size δ is sufficiently large. We find that the solution of the peridynamic model converges to that of the fractional Laplacian model at a rate of O(δ −α). 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.
In this paper, we propose an accurate finite difference method to discretize the d-dimensional (f... more In this paper, we propose an accurate finite difference method to discretize the d-dimensional (for d ≥ 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 ε), for u ∈ C 0, α+ε (Ω) if α < 1 (or u ∈ C 1, α−1+ε (Ω) if α ≥ 1) with ε > 0, suggesting the minimum consistency conditions. The accuracy can be improved to O(h 2), for u ∈ C 2, α+ε (Ω) if α < 1 (or u ∈ C 3, α−1+ε (Ω) if α ≥ 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 ∈ C 1,1 (Ω) for any α ∈ (0, 2). Moreover, if the solution of tempered fractional Poisson problems satisfies u ∈ C p,s (Ω) for p = 0, 1 and s ∈ (0, 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.
Computer Methods in Applied Mechanics and Engineering, 2019
In this paper, we propose accurate and efficient finite difference methods to discretize the twoa... more In this paper, we propose accurate and efficient finite difference methods to discretize the twoand three-dimensional fractional Laplacian (−∆) α 2 (0 < α < 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 α → 2 − , they collapse to the central difference schemes of the classical Laplace operator −∆. 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 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 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.
We study a space-fractional Allen-Cahn phase-field model in which the fractional Laplacian is def... more We study a space-fractional Allen-Cahn phase-field model in which the fractional Laplacian is defined to be an infinitesimal generator of an α-stable Lévy process with 0 < α < 2. Compared with the Allen-Cahn model using the Riemann-Liouville (R-L) fractional derivatives in the literature, the current model enjoys the following advantages: (1) it can control the sharpness and the decaying behavior of the interface via the tunable order parameter α as the Riemann-Liouville fractional model does; (2) it generates numerical solutions with sharper interfaces than the Riemann-Liouville fractional Allen-Cahn model with the same 1 < α < 2, and it can generate numerical solutions with even sharper interfaces as it also applies to the case for 0 < α < 1; (3) as α → 2 − , it provides good approximations to the classical Allen-Cahn model as the fractional Riemann-Liouville model does; (4) it is rotational invariant; and (5) it has a much slower mass decaying rate than the Riemann-Liouville fractional model. We also propose and analyze a first order in time and second order in space energy stable finite difference scheme.
In this paper, we develop a novel finite difference method to discretize the fractional Laplacian... more In this paper, we develop a novel finite difference method to discretize the fractional Laplacian (−) 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 ∈ C 3,α/2 (R), our method has an accuracy of O(h 2) uniformly for any α ∈ (0, 2), while for u ∈ C 1,α/2 (R), the accuracy is O(h 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.
Computers & Mathematics with Applications, 2017
In this paper, we propose a fast algorithm for efficient and accurate solution of the space-time ... more 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 (−∆) α/2 s 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.
Computers & Mathematics with Applications, 2016
We propose three Fourier spectral methods, i.e., the split-step Fourier spectral (SSFS), the Cran... more 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 secondorder 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.
In this paper, we numerically study the ground and first excited states of the fractional Schrödi... more In this paper, we numerically study the ground and first excited states of the fractional Schrödinger equation in an infinite potential well. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrödinger equation analytically. We first introduce a normalized fractional gradient flow and then discretize it by a quadrature rule method in space and the semi-implicit Euler method in time. Our numerical results suggest that the eigenfunctions of the fractional Schrödinger equation in an infinite potential well differ from those of the standard (non-fractional) Schrödinger equation. 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 ...
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