Table of Content

20 June 2020, Volume 2 Issue 2

Previous Issue    Next Issue

Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study

Bangti Jin, Zhi Zhou

2020, 2(2):  163-177.  doi:10.1007/s42967-019-00042-9

Asbtract ( 8957 )   PDF  

References | Related Articles | Metrics

In this work, we develop an efcient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of order α(0, 1) in time. The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear fnite elements in space and corrected high-order BDF convolution quadrature in time. At each time step, instead of solving the linear algebraic system exactly, we employ a multigrid iteration with a Gauss-Seidel smoother to approximate the solution efciently. Illustrative numerical results for nonsmooth problem data are presented to demonstrate the app

A High-Order Scheme for Fractional Ordinary Diferential Equations with the Caputo-Fabrizio Derivative

Junying Cao, Ziqiang Wang, Chuanju Xu

2020, 2(2):  179-199.  doi:10.1007/s42967-019-00043-8

Asbtract ( 754 )   PDF  

References | Related Articles | Metrics

In this paper, we consider numerical solutions of fractional ordinary diferential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach.

Nonlocal Dynamics for Non-Gaussian Systems Arising in Biophysical Modeling

Xiaoli Chen, Jinqiao Duan

2020, 2(2):  201-213.  doi:10.1007/s42967-019-00046-5

Asbtract ( 2001 )   PDF  

References | Related Articles | Metrics

The aim of this article is to review our recent work on nonlocal dynamics of non-Gaussian systems arising in a gene regulatory network. We have used the mean exit time, escape probability and maximal likely trajectory to quantify dynamical behaviors of a stochastic diferential system with non-Gaussian α-stable Lévy motions, to examine how the nonGaussianity index and noise intensity afect the gene transcription processes.

An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients

Xue-lei Lin, Pin Lyu, Michael K. Ng, Hai-Wei Sun, Seakweng Vong

2020, 2(2):  215-239.  doi:10.1007/s42967-019-00050-9

Asbtract ( 7839 )   PDF  

References | Related Articles | Metrics

In this paper, a second-order fnite-diference scheme is investigated for time-dependent space fractional difusion equations with variable coefcients. In the presented scheme, the Crank-Nicolson temporal discretization and a second-order weighted-and-shifted Grünwald-Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the variable coefcients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes, so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efciency of the proposed scheme.

Numerical Computations of Nonlocal Schrödinger Equations on the Real Line

Yonggui Yan, Jiwei Zhang, Chunxiong Zheng

2020, 2(2):  241-260.  doi:10.1007/s42967-019-00052-7

Asbtract ( 688 )   PDF  

References | Related Articles | Metrics

The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artifcial boundary method, we frst derive the exact artifcial nonrefecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in Tian and Du (SIAM J Numer Anal 51:3458-3482, 2013) to discretize the nonlocal operator, and apply the z-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonrefecting boundary condition and leads us to reformulate the original infnite discrete system into an equivalent fnite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are fnally provided to demonstrate the efectiveness of our approach.

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

Burak Aksoylu, Fatih Celiker, George A. Gazonas

2020, 2(2):  261-303.  doi:10.1007/s42967-019-00051-8

Asbtract ( 11547 )   PDF  

References | Related Articles | Metrics

We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics, a nonlocal formulation of continuum mechanics. We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation. A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero. We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible. We fnd that, for polynomial degrees greater than or equal to two, there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible. We verify these fndings through extensive numerical experiments.

Jacobi Collocation Methods for Solving Generalized Space-Fractional Burgers' Equations

Qingqing Wu, Xiaoyan Zeng

2020, 2(2):  305-318.  doi:10.1007/s42967-019-00053-6

Asbtract ( 725 )   PDF  

References | Related Articles | Metrics

The aim of this paper is to obtain the numerical solutions of generalized space-fractional Burgers' equations with initial-boundary conditions by the Jacobi spectral collocation method using the shifted Jacobi-Gauss-Lobatto collocation points. By means of the simplifed Jacobi operational matrix, we produce the diferentiation matrix and transfer the space-fractional Burgers' equation into a system of ordinary diferential equations that can be solved by the fourth-order Runge-Kutta method. The numerical simulations indicate that the Jacobi spectral collocation method is highly accurate and fast convergent for the generalized space-fractional Burgers' equation.