【作者】 李娟；

【导师】 游雄；

【作者基本信息】 南京农业大学 ， 应用数学， 2016， 硕士

【摘要】 振荡微分方程在分子动力学、天文学、生物学等科学与工程应用领域广泛存在并起着重要的作用.然而大部分振荡微分方程的解析解是很难得到的,因此有效的数值方法是研究振荡微分方程的重要途径.分裂方法是数值求解振荡问题的一种非常重要而且有效的方法,它主要针对可以通过某种方法分裂为若干个可积的子系统的复杂系统.由于分裂方法形式简单,通常为显式格式便于编程实现,并且能有效的保持系统的定性性质.近年来,分裂方法引起越来越多的学者的兴趣.本文旨在建立能够保持微分方程解的振荡性质的分裂方法,并应用于生物振子的数值模拟.本文共分为四章.第一章为预备知识,主要概述了研究背景和微分方程数值方法的基本概念.引入了一阶Lie-Trotter分裂法、二阶Strang分裂法、基于Strang分裂法的三级四阶三级跳,对一般形式的分裂方法给出了阶条件.介绍了两类生物振荡问题(食物-捕食者系统与基因调控网络)的微分方程模型.第二章分析了分裂方法的相性质,包括色散和误差.证明了Lie-Totter分裂法和Strang分裂法有2阶色散,基于Strang分裂的三级跳有4阶色散,三个分裂法都是零耗散的.在此基础上构造了相拟合的Lie-Trotter分裂法、Strang分裂法和三级跳,并分析了它们的数值稳定性.数值试验显示了新的相拟合分裂法比传统的分裂法及相同代数阶的Runge-Kutta方法更为高效.第三章运用指数变换以及冻结的技巧构造了指数型Lie-Trotter分裂方法及指数型Strang分裂方法,通过数值实验验证了指数型分裂法的优越性.第四章在传统的Runge-Kutta方法的基础上结合分裂思想,讨论了分裂的Runge-Kutta方法.利用给出的阶条件构造了二阶、三阶分裂Runge-Kutta方法.数值试验表明新方法比同阶的RK方法具有更高的效率.最后,简要的总结了本文的主要研究贡献,并对以后的研究工作提出了展望.更多还原

【Abstract】 Oscillatory differential equations play an important role in a wide variety of applied fields in science and engineering,such as molecular dynamics,astronomy,biology and so on.However,it is difficult in general to obtain the analytic solution to a differential equation.Therefore,effective numerical solution will be the significant way to investigate the behaviors of oscillatory differential equations.Splitting methods form one of the most important and powerful tools for numerical investigation of complicated systems which can be decomposed into some subsystems in an appropriate way.Despite the simple forms,splitting methods,usually explicit,easy to be implemented,can preserve qualitative properties of the system.In the recent years,splitting methods have attracted more and more interest.The purpose of this thesis is to develop splitting methods which can preserve the oscillatory properties of the solution of differential equations.Furthermore,the newly constructed splitting integrators will be applied to the numerical simulation of biological oscillators.The thesis is divided into four chapters.In Chapter 1,as the preliminary of the thesis,we present the background and elementary concepts of numerical methods for ordinary differential equations.The Lie-Trotter splitting of order one,Strang splitting of order two and Strang-splitting-based triple jump are introduced.The order conditions for general splitting methods are present.Two biological oscillators-the prey-predator system and the gene regulatory networks-are modeled in differential equations.Chapter 2 is devoted to the phase properties-dispersion and dissipation-of splitting methods.We prove that the Lie-Trotter splitting and the Strang splitting are dispersive of order two and the Strang-splitting-based triple jump is dispersive of order four.All the three splitting methods are zero-dissipative.We further derive the phase fitted Lie-Trotter split-ting,Strang splitting and triple jump.Numerical stability of the new methods is analyzed.The numerical results show that the new phase fitted methods are more efficient than their prototype splitting methods and Runge-Kutta methods of the same algebraic order.In chapter 3,exponential transformation and the time-freezing technique are utilized to construct the exponential Lie-Trotter splitting method and the exponential Strang splitting method.Their superiority is verified by numerical experiments.In chapter 4,based on the idea of blending the classical Runge-Kutta methods with splitting,the splitting Runge-Kutta methods are developed and investigated.The order conditions up to order three are presented.Two second-order and one third-order splitting Runge-Kutta methods are derived.Numerical results show that the new methods are much more efficient than the existed RK methods with the same order.Finally,we summarize the main contributions of this thesis and present the prospect for the future work.更多还原