耦合非线性薛定谔方程组孤立子解的局部间断Petrov-Galerkin方法数值模拟  

Local Discontinuous Petrov-Galerkin Method for Simulating Solitary Waves of the Coupled Nonlinear Schrodinger Equations

作　　者：赵国忠[1] 蔚喜军[2] ZHAO Guozhong;YU Xijun(School of Mathematical Sciences,Baotou Teachers’College,Baotou 014030;Key Laboratory of Computational Physics,Institute of Applied Physics and Computational Mathematics,Beijing 100088)

机构地区：[1]包头师范学院数学科学学院,包头014030 [2]北京应用物理与计算数学研究所计算物理重点实验室,北京100088

出　　处：《工程数学学报》2024年第6期1109-1132,共24页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(12361076,11761054,12071046,11261035);内蒙古自治区自然科学基金(2021MS01001,2015MS0108,2012MS0102);内蒙古自治区高等学校科学研究项目基金(NJZY22036,NJZY19186,NJZZ12198).

摘　　要：耦合非线性薛定谔方程组在量子物理、非线性光学、晶体物理、波色–爱因斯坦凝聚和水波动力学等很多物理领域有着重要的应用价值。提出了一种局部间断PetrovGalerkin方法。首先,将耦合非线性薛定谔方程组改写为一阶微分方程组。空间离散采用间断Petrov-Galerkin方法,时间离散采用三阶总变差不增Runge-Kutta方法。数值实验表明,该算法对线性元和二次元都能达到最优收敛阶。通过数值算例计算了质量、动量和能量守恒量,该算法可以很好地模拟单孤立子传输、双孤立子碰撞和三孤立子碰撞现象。此外,该算法可以在较长的时间间隔内模拟复杂波型的相互作用或传播,还可以模拟孤子传输和孤子产生现象。The coupled nonlinear Schrodinger equations have important applications in many areas of physics,such as quantum physics,nonlinear optics,crystal physics,Bose-Einstein condensates,water wave dynamics and so on.In this paper,a local discontinuous Petrov-Galerkin method is developed.The coupled nonlinear Schrodinger equations are firstly rewritten as a first order differential system.The spacial discretization is accomplished by the discontinuous Petrov-Galerkin method.The third order total variation diminishing Runge-Kutta method is used to finish the temporal discretization.The numerical experiments shown that the algorithm can reach its optimal convergence order for both linear and quadratic elements.The mass,momentum and energy conservation quantities are evaluated by some numerical examples.The algorithm can simulate the single soliton propagation,double soliton collision and three soliton collision phenomena very well.In addition,the algorithm can simulate the complex wave interaction or propagation in a long time interval.Furthermore,the scheme can simulate the soliton transmission and soliton creation phenomena.

关 键 词：局部间断Petrov-Galerkin方法 耦合非线性薛定谔方程 孤立子碰撞 守恒量 

分 类 号：O241.82[理学—计算数学]