scholarly journals Fourier Spectral Method for a Class of Nonlinear Schrödinger Models

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Lei Zhang ◽  
Weihua Ou Yang ◽  
Xuan Liu ◽  
Haidong Qu
Keyword(s):  
Spectral Method ◽  
Fourth Order ◽  
Fourier Spectral Method ◽  
Spatial Direction ◽  
Exponential Time ◽  
Exponential Time Differencing ◽  
Nonlinear Schrödinger ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Temporal Direction

In this paper, Fourier spectral method combined with modified fourth order exponential time-differencing Runge-Kutta is proposed to solve the nonlinear Schrödinger equation with a source term. The Fourier spectral method is applied to approximate the spatial direction, and fourth order exponential time-differencing Runge-Kutta method is used to discrete temporal direction. The proof of the conservation law of the mass and the energy for the semidiscrete and full-discrete Fourier spectral scheme is given. The error of the semidiscrete Fourier spectral scheme is analyzed in the proper Sobolev space. Finally, several numerical examples are presented to support our analysis.

Numerical simulation of a class of space fractional bistable systems based on the Fourier spectral method

2021 ◽  
pp. 146134842110637
Author(s):  
Haitao Liu ◽  
Wang Yulan ◽  
Li Cao ◽  
Wei Zhang
Keyword(s):  
Spectral Method ◽  
Fractional Order ◽  
Fourth Order ◽  
Numerical Approach ◽  
Bistable System ◽  
Fourier Spectral Method ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Bistable Systems ◽  
Runge Kutta

Nonlinear vibration arises everywhere in a bistable system. The bistable system has been widely applied in physics, biology, and chemistry. In this article, in order to numerically simulate a class of space fractional-order bistable system, we introduce a numerical approach based on the modified Fourier spectral method and fourth-order Runge-Kutta method. The fourth-order Runge-Kutta method is used in time, and the Fourier spectrum is used in space to approximate the solution of the space fractional-order bistable system. Numerical experiments are given to illustrate the effectiveness of this method.

scholarly journals Fourth Order Exponential Time Differencing Method with Local Discontinuous Galerkin Approximation for Coupled Nonlinear Schrödinger Equations

2015 ◽  
Vol 17 (2) ◽  
pp. 510-541 ◽  
Author(s):  
X. Liang ◽  
A. Q. M. Khaliq ◽  
Y. Xing
Keyword(s):  
Discontinuous Galerkin ◽  
Fourth Order ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Exponential Time ◽  
Exponential Time Differencing ◽  
Nonlinear Schrödinger ◽  
Local Discontinuous Galerkin ◽  
Nonlinear Schrodinger Equations

AbstractThis paper studies a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

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