Yanzhi Zhanghttps://works.bepress.com/yanzhi-zhang/Recent works by Yanzhi Zhangen-usCopyright (c) 2019 All rights reserved.Tue, 01 Oct 2019 07:00:00 +00003600Accurate Numerical Methods for Two and Three Dimensional Integral Fractional Laplacian with Applicationshttps://works.bepress.com/yanzhi-zhang/32/<p>In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian (-Δ)<sup>α/2</sup> (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<sup>-</sup>, they collapse to the central difference schemes of the classical Laplace operator −Δ. We prove that our methods are consistent if 𝑢 ∈ C<sup>[α], α –[α]+ ε </sup>(ℝ<sup>d</sup>), and the local truncation error is 𝓞 (h<sup>ε</sup>), with ε > 0 a small constant and [· } denoting the floor function. If 𝑢 ∈ C<sup>2+[α], α -[ α]+ ε </sup>(ℝ<sup>d</sup>), 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 <em>at most</em> satisfy 𝑢 ∈ C<sup>1,1</sup> (ℝ<sup>d</sup>). 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.</p>
Tue, 01 Oct 2019 07:00:00 +0000https://works.bepress.com/yanzhi-zhang/32/Research WorksA Comparative Study on Nonlocal Diffusion Operators Related to the Fractional Laplacianhttps://works.bepress.com/yanzhi-zhang/26/<p>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-<sup>a</sup>). 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.</p>
Tue, 01 Jan 2019 08:00:00 +0000https://works.bepress.com/yanzhi-zhang/26/Research WorksNumerical Investigation of Dynamics of Elliptical Magnetic Microparticles in Shear Flowshttps://works.bepress.com/yanzhi-zhang/28/<p>We study the rotational dynamics of magnetic prolate elliptical particles in a simple shear flow subjected to a uniform magnetic field, using direct numerical simulations based on the finite element method. Focusing on paramagnetic and ferromagnetic particles, we investigate the effects of the magnetic field strength and direction on their rotational dynamics. In the weak field regime (below a critical field strength), the particles are able to perform complete rotations, and the symmetry property of particle rotational speed is influenced by the direction and strength of the magnetic field. In the strong field regime (above a critical strength), the particles are pinned at steady angles. The steady angle depends on both the direction and strength of the magnetic field. Our results show that paramagnetic and ferromagnetic particles exhibit markedly different rotational dynamics in a uniform magnetic field. The numerical findings are in good agreement with theoretical prediction. Our numerical investigation further reveals drastically different lateral migration behaviors of paramagnetic and ferromagnetic particles in a wall-bounded simple shear flow under a uniform magnetic field. These two kinds of particles can thus be separated by combining a shear flow and a uniform magnetic field. We also study the lateral migration of paramagnetic and ferromagnetic particles in a pressure-driven flow (a more practical flow configuration in microfluidics), and observe similar lateral migration behaviors. These findings demonstrate a simple but useful way to manipulate non-spherical microparticles in microfluidic devices.</p>
Wed, 01 Aug 2018 07:00:00 +0000https://works.bepress.com/yanzhi-zhang/28/Research WorksDynamics of Paramagnetic and Ferromagnetic Ellipsoidal Particles in Shear Flow under a Uniform Magnetic Fieldhttps://works.bepress.com/yanzhi-zhang/29/<p>We investigate the two-dimensional dynamic motion of magnetic particles of ellipsoidal shapes in shear flow under the influence of a uniform magnetic field. In the first part, we present a theoretical analysis of the rotational dynamics of the particles in simple shear flow. By considering paramagnetic and ferromagnetic particles, we study the effects of the direction and strength of the magnetic field on the particle rotation. The critical magnetic-field strength, at which particle rotation is impeded, is determined. In a weak-field regime (i.e., below the critical strength) where the particles execute complete rotations, the symmetry property of the rotational velocity is shown to depend on the direction of the magnetic field. In a strong-field regime (i.e., above the critical strength), the particles are impeded at steady angles and the stability of these angles is examined. Under a uniform field, paramagnetic and ferromagnetic particles behave differently, in terms of the critical strength, symmetry property of the rotational velocity, and steady angles. In the second part, we use two-dimensional numerical simulations to study the implications of rotational dynamics for lateral migration of the particles in wall-bound shear flows. In the weak-field regime, the paramagnetic prolate particles migrate away when the field is applied perpendicular to the flow and towards the bounded wall when the field is applied parallel to the flow. Ferromagnetic particles exhibit negligible migration under fields that are parallel or perpendicular to the flow. The different lateral migration behaviors are due to the difference in the symmetry property of particle rotational velocity. In the strong-field regime, the particles are impeded at different stable steady angles, which result in different lateral migration behaviors as well. The fundamental insights from our work demonstrate various feasible strategies for manipulating paramagnetic and ferromagnetic particles.</p>
Wed, 01 Aug 2018 07:00:00 +0000https://works.bepress.com/yanzhi-zhang/29/Research WorksA Fast Algorithm for Solving the Space-Time Fractional Diffusion Equationhttps://works.bepress.com/yanzhi-zhang/27/<p>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)<sub>s</sub><sup>α/2</sup> 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.</p>
Thu, 01 Mar 2018 08:00:00 +0000https://works.bepress.com/yanzhi-zhang/27/Research WorksA Novel and Accurate Finite Difference Method for the Fractional Laplacian and the Fractional Poisson Problemhttps://works.bepress.com/yanzhi-zhang/30/<p>In this paper, we develop a novel finite difference method to discretize the fractional Laplacian (−Δ)<sup>α/2</sup> 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<sup>3,α/2</sup>(R), our method has an accuracy of O(h<sup>2</sup>) uniformly for any α ∈ (0,2), while for u ∈ C<sup>1,α/2</sup>(R), the accuracy is O(<sup>1-α/2</sup>). 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.</p>
Thu, 01 Feb 2018 08:00:00 +0000https://works.bepress.com/yanzhi-zhang/30/Research WorksMagnetic Control of Lateral Migration of Ellipsoidal Microparticles in Microscale Flowshttps://works.bepress.com/yanzhi-zhang/25/Tue, 01 Aug 2017 07:00:00 +0000https://works.bepress.com/yanzhi-zhang/25/Research WorksConvergence of the Spectral Galerkin Method for the Stochastic Reaction–Diffusion–Advection Equationhttps://works.bepress.com/yanzhi-zhang/2/<p>We study the convergence of the spectral Galerkin method in solving the stochastic reaction-diffusion-advection equation under different Lipschitz conditions of the reaction function f. When f is globally (locally) Lipschitz continuous, we prove that the spectral Galerkin approximation strongly (weakly) converges to the mild solution of the stochastic reaction–diffusion–advection equation, and the rate of convergence in H<sub>r</sub>-norm is (1/2−r)<sup>-</sup>, for any r ∈ [0, 1/2) (r ∈ (1/2 – 1/2d ,1/2)). The convergence analysis in the local Lipschitz case is challenging, especially in the presence of an advection term. We propose a new approach based on the truncation techniques, which can be easily applied to study other stochastic partial differential equations. Numerical simulations are also provided to study the convergence of Galerkin approximations.</p>
Wed, 15 Feb 2017 08:00:00 +0000https://works.bepress.com/yanzhi-zhang/2/Research WorksFractional Schrödinger Dynamics and Decoherencehttps://works.bepress.com/yanzhi-zhang/14/<p>We study the dynamics of the Schrödinger equation with a fractional Laplacian (−Δ)<sup>α</sup> and the decoherence of the solution is observed. Analytically, we obtain equations of motion for the expected position and momentum in the fractional Schödinger equation, equations that are the fractional counterpart of the well-known Newtonian equations of motion for the standard (α=1α=1) Schrödinger equation. Numerically, we propose an explicit, effective numerical method for solving the time-dependent fractional nonlinear Schrödinger equation—a method that has high order spatial accuracy, requires little memory, and has low computational cost. We apply our method to study the dynamics of fractional Schrödinger equation and find that the nonlocal interactions from the fractional Laplacian introduce decoherence into the solution. The local nonlinear interactions can however reduce or delay the emergence of decoherence. Moreover, we find that the solution of the standard NLS behaves more like a particle, but the solution of the fractional NLS behaves more like a wave with interference effects.</p>
Sat, 01 Oct 2016 07:00:00 +0000https://works.bepress.com/yanzhi-zhang/14/Research WorksMass Conservative Method for Solving the Fractional Nonlinear Schrödinger Equationhttps://works.bepress.com/yanzhi-zhang/18/<p>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.</p>
Wed, 01 Jun 2016 07:00:00 +0000https://works.bepress.com/yanzhi-zhang/18/Research Works