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Breather and solitons waves in optical fibers via exponential time differencing method

COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2020年 85卷 105237-105237页

作者： Ashi, Hala A. Aljahdaly, Noufe H. King Abdulaziz Univ Fac Sci Dept Math Jeddah 21589 Saudi Arabia King Abdulaziz Univ Dept Math Fac Sci & Arts Rabigh Campus Jeddah Saudi Arabia

This work proposes the numerical solutions of the cubic nonlinear Schrödinger equation subjected to rational initial value for three cases, (i) Akhmediev breather (ii) Peregrine and (iii) Kuznetsov-Ma solitons by operating exponential time differencing method (ETDM). The work shows that ETDM is a reliable and accurate method to find the solutions comparing to Adomian decomposition method (ADM) and Laplace-Adomian decomposition method (LADM).

关键词： exponential time differencing method Nonlinear Schrödinger equation Peregrine soliton Akhmediev breather Kuznetsov-Ma soliton

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Efficient Krylov-based exponential time differencing method in application to 3D advection-diffusion-reaction systems

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APPLIED MATHEMATICS AND COMPUTATION 2018年 338卷 260-273页

作者： Bhatt, H. P. Khaliq, A. Q. M. Wade, B. A. Savannah State Univ Dept Math Savannah GA 31404 USA Middle Tennessee State Univ Dept Math Sci Murfreesboro TN 37132 USA Middle Tennessee State Univ Ctr Computat Sci Murfreesboro TN 37132 USA Univ Wisconsin Dept Math Sci Milwaukee WI 53201 USA

The number of ordinary differential equations generally increases exponentially as the partial differential equation is posed on a domain with more dimensions. This is, of course, the curse of dimensionality for exponential time differencing methods. The computational challenge in applying exponential time differencing methods for solving partial differential equations in high spatial dimensions is how to compute the matrix exponential functions for very large matrices accurately and efficiently. In this paper, our main aim is to design a Krylov subspace approximation-based locally extrapolated exponential time differencing method and compare its performance in terms of accuracy and efficiency to the already available method in the literature for solving a three-dimensional nonlinear advectiondiffusion-reaction systems. The fundamental idea of the proposed method is to compute only the action of the matrix exponential on a given state vector instead of computing the matrix exponential itself, and then multiplying it with given vector. The stability and local truncation error of the proposed method have been examined. Calculation of local truncation error and empirical convergence analysis indicate the proposed method is second-order accurate in time. The performance and reliability of this novel method have been investigated by testing it on systems of the three-dimensional nonlinear advection-diffusionreaction equations and three-dimensional viscous nonlinear Burgers' equation. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Krylov subspace approximation Local extrapolation exponential time differencing method Advection-diffusion-reaction equations Burgers' equation

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AN ENERGY STABLE AND MAXIMUM BOUND PRINCIPLE PRESERVING SCHEME FOR THE DYNAMIC GINZBURG--LANDAU EQUATIONS UNDER THE TEMPORAL GAUGE\ast

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2023年第6期61卷 2695-2717页

作者： Ma, Limin Qiao, Zhonghua Wuhan Univ Sch Math & Stat Wuhan 430072 Hubei Peoples R China Hong Kong Polytech Univ Dept Appl Math Hung Hom Kowloon Hong Kong Peoples R China

This paper proposes a decoupled numerical scheme of the time-dependent Ginzburg-Landau equations under the temporal gauge. For the magnetic potential and the order parameter, the discrete scheme adopts the second type Ned\'elec element and the linear element for spatial discretization, respectively;and a linearized backward Euler method and the first order exponential time differencing method for time discretization, respectively. The maximum bound principle (MBP) of the order parameter and the energy dissipation law in the discrete sense are proved. The discrete energy stability and MBP preservation can guarantee the stability and validity of the numerical simulations, and further facilitate the adoption of an adaptive time-stepping strategy, which often plays an important role in long-time simulations of vortex dynamics, especially when the applied magnetic field is strong. An optimal error estimate of the proposed scheme is also given. Numerical examples verify the theoretical results of the proposed scheme and demonstrate the vortex motions of superconductors in an external magnetic field.

关键词： Ginzburg--Landau equations energy stability maximum bound principle error estimate exponential time differencing method

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exponential integrator preserving mass boundedness and energy conservation for nonlinear Schrodinger equation

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APPLIED NUMERICAL MATHEMATICS 2022年 173卷 308-328页

作者： Xu, Zhuangzhi Cai, Wenjun Hu, Dongdong Wang, Yushun Henan Polytech Univ Sch Math & Informat Sci Jiaozuo Henan Peoples R China Nanjing Normal Univ Jiangsu Collaborat Innovat Ctr Biomed Funct Mat Sch Math Sci Jiangsu Key Lab NSLSCS Nanjing 210023 Peoples R China

In this paper, a novel linearly implicit conservative scheme is developed for the two-dimensional nonlinear Schrodinger equation. The scheme is based on a new linearly implicit exponential time differencing method for the temporal discretization and the Fourier pseudo-spectral approximation for the spatial derivative. By rigorous analysis, we prove that the proposed scheme can preserve the energy conservation. Furthermore, although in fact the proposed scheme does not preserve mass conservation, we can prove that it can preserve the discrete L-2 norm boundedness of the numerical solution, which may be helpful for the analysis of unconditional convergence of the energy conservative scheme. Then an optimal error estimate of the proposed scheme is established without any restriction on the grid ration. Numerical experiments show that the proposed scheme is remarkable efficiency comparing with some other existing structure-preserving schemes. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Nonlinear Schrodinger equation Linearly implicit exponential time differencing method Fourier pseudo-spectral method Energy conservation Error estimate

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An accurate and efficient numerical method for neural field models with transmission delays

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2022年 135卷 206-216页

作者： Zhao, W. Hon, Y. C. Guangdong Univ Technol Sch Appl Math Guangzhou Peoples R China City Univ Hong Kong Dept Math Hong Kong Peoples R China

Based on the recently developed localized radial basis function collocation method (LRBFCM) and exponential time differencing (ETD) method, we derive in this paper an accurate and efficient numerical method for solving neural field models with space-dependent delays formulated as system of integro-differential equations. This combined LRBFCM-ETD method inherits the advantages of these two methods in space and time, thereby offering high accuracy and efficiency for both uniform and non-uniform discretizations, extending the applicability of the method to multi-dimensions easily, and maintaining a good behaviour in a large time-step and long-time integration. Some numerical experiments are presented and numerical results show that the obtained simulations via the LRBFCM-ETD method are reliable and effective for approximating the solution of models investigated in the current paper.

关键词： Integro-differential equations Space-dependent delays Localized radial basis function collocation method exponential time differencing method

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exponential time differencing Algorithm for Pulse-Coupled Hodgkin-Huxley Neural Networks

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FRONTIERS IN COMPUTATIONAL NEUROSCIENCE 2020年 14卷 40页

作者： Tian, Zhong-qi Kyle Zhou, Douglas Shanghai Jiao Tong Univ Sch Math Sci Inst Nat Sci MOE LSC Shanghai Peoples R China

The exponential time differencing (ETD) method allows using a large time step to efficiently evolve stiff systems such as Hodgkin-Huxley (HH) neural networks. For pulse-coupled HH networks, the synaptic spike times cannot be predetermined and are convoluted with neuron's trajectory itself. This presents a challenging issue for the design of an efficient numerical simulation algorithm. The stiffness in the HH equations are quite different, for example, between the spike and non-spike regions. Here, we design a second-order adaptive exponential time differencing algorithm (AETD2) for the numerical evolution of HH neural networks. Compared with the regular second-order Runge-Kutta method (RK2), our AETD2 method can use time steps one order of magnitude larger and improve computational efficiency more than ten times while excellently capturing accurate traces of membrane potentials of HH neurons. This high accuracy and efficiency can be robustly obtained and do not depend on the dynamical regimes, connectivity structure or the network size.

关键词： Hodgkin-Huxley exponential time differencing method efficiency pulse-coupled second-order

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Dynamical study of two predators and one prey system with fractional Fourier transform method

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NUMERICAL methodS FOR PARTIAL DIFFERENTIAL EQUATIONS 2018年第5期34卷 1614-1636页

作者： Owolabi, Kolade M. Pindza, Edson Davison, Matt Univ Free State Inst Groundwater Studies Fac Nat & Agr Sci ZA-9300 Bloemfontein South Africa Univ Western Ontario Dept Stat & Actuarial Sci London ON N6A 5B7 Canada

In this work, we investigate both the analytical and numerical studies of the dynamical model comprising of three species systems. We analyze the linear stability of stationary solutions in the one-dimensional multisystem modeling the interactions of two predators and one prey species. The stability analysis has a lot of implications for understanding the various spatiotemporal and chaotic behaviors of the species in the spatial domain. The analysis results presented have established the possibility of the three-interacting species to coexist harmoniously, this feat is achieved by combining the local and global analyses to determine the global dynamics of the system. In the presence of a fractional diffusion term, we introduced a fractional Fourier transform for solving the system modeled by fractional partial differential equations. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator with exponential accuracy and a completely straightforward extension to high spatial dimensions. The scheme is described in detail and justified by a number of computational experiments.

关键词： coexistence exponential time differencing method fractional Fourier transform global and local stability nonlinear predator-prey model reaction-diffusion system

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exponential time differencing algorithm for Debye medium in FDTD

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INTERNATIONAL JOURNAL OF INFRARED AND MILLIMETER WAVES 2007年第11期28卷 1025-1031页

作者： Huang, Sh. J. Zhanjinag Normal Univ Dept Phys Zhanjiang 524048 Guangdong Peoples R China

The exponential time differencing (ETD) formultion is proposed for Debye medium using finite-difference time-domain (FDTD) method. The schemes of the auxiliary differential equation (ADE) and ETD algorithms are presented. The algorithms are validated by comparing the simulation results with analytical values. The ETD method costs the same memory as the ADE method, and the numerical results indicate that the ETD technique is more accurate than the ADE method.

关键词： finite-difference time-domain method exponential time differencing method auxiliary differential equation method dispersive Debye medium

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Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations

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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2006年第25-28期195卷 3430-3447页

作者： Xu, Yan Shu, Chi-Wang Univ Sci & Technol China Dept Math Hefei 230026 Anhui Peoples R China Brown Univ Div Appl Math Providence RI 02912 USA

In this paper we develop a local discontinuous Galerkin method to solve the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. The L-2 stability of the schemes is obtained for both of these nonlinear equations. We use both the traditional nonlinearly stable explicit high order Runge-Kutta methods and the explicit exponential time differencing method for the time discretization;the latter can achieve high order accuracy and maintain good stability while avoiding the very restrictive explicit stability limit of the former when the PDE contains higher order spatial derivatives. Numerical examples are shown to demonstrate the accuracy and capability of these methods. (c) 2005 Elsevier B.V. All rights reserved.

关键词： local discontinuous Galerkin method Kuramoto-Sivashinsky equation Ito-type coupled KdV equations stability exponential time differencing method

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Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations

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PHYSICA D-NONLINEAR PHENOMENA 2005年第1-2期208卷 21-58页

作者： Xu, Y Shu, CW Brown Univ Div Appl Math Providence RI 02912 USA Univ Sci & Technol China Dept Math Hefei 230026 Anhui Peoples R China

In this paper, we develop, analyze and test local discontinuous Galerkin (DG) methods to solve two classes of two-dimensional nonlinear wave equations formulated by the Kadomtsev-Petviashvili (KP) equation and the Zakharov-Kuznetsov (ZK) equation. Our proposed scheme for the Kadomtsev-Petviashvili equation satisfies the constraint from the PDE which contains a non-local operator and at the same time has the local property of the discontinuous Galerkin methods. The scheme for the Zakharov-Kuznetsov equation extends the previous work on local discontinuous Galerkin method solving one-dimensional nonlinear wave equations to the two-dimensional case. L-2 stability of the schemes is proved for both of these two nonlinear equations. Numerical examples are shown to illustrate the accuracy and capability of the methods. (c) 2005 Elsevier B.V. All rights reserved.

关键词： local discontinuous Galerkin method Kadomtsev-Petviashvili equation Zakharov-Kuznetsov equation stability exponential time differencing method

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