二阶振荡守恒/耗散系统的几何数值积分
发布时间：2021-07-24 15:44
本论文研究下述多频高维振荡二阶微分方程系统的数值积分。其中M∈Rd×d是一个半正定矩阵,隐含了该系统的主振荡频率,y∈Rd,f一方面,如果（1）中右端项函数f依赖于y’,系统为一个阻尼系统,其能量是耗散的。另一方面,如果（1）中M是一个对称半正定矩阵且f（y）是实值函数U（y）的负梯度,那么,这就是一个哈密尔顿系统。此时系统有两个重要性质：辛性和能量守恒。在数值计算领域人们已经认识到,设计数值算法应当考虑所研究问题的特殊结构。近年来,几何数值积分得到了越来越多的关注。一个数值方法是几何算法如果方法能够精确保持系统的一个或多个物理／几何结构。更确切的说,为了使得数值算法能够更正确反映系统的定性行为,要尽量保证原系统的基本结构不会被数值算法破坏。因此,我们重点关注系统（1）的几何数值积分,使得算法保持原系统尽可能多的结构特性。系统（1）的显著特性：振荡性的保持将贯穿全文。本论文第一部分的研究考虑振荡耗散系统。对于带阻尼（含有y’）的振荡二阶常微分方程初值问题,我们研究了高维ARKN方法的构造。同时,对于求解右端函数项依赖y和y’的一般振荡二阶常微分方程的数值方法,通过引入新的线性测试模型,...
【文章来源】：南京大学江苏省 211工程院校 985工程院校 教育部直属院校
【文章页数】：163 页
【学位级别】：博士
【文章目录】：
Acknowledgements
摘要
Abstract
Chapter 1 Introduction
1.1 Numerical methods for oscillatory problems
1.2 Conservative/dissipative systems and geometric numerical integration
1.3 Layout of thesis
Chapter 2 Multi-frequency and multdimensionalARKN methods for general oscillatorysecond-order initial value problems
2.1 Multi-frequency and multidimensional ARKN methods and the corre-sponding order conditions
2.2 Novel multi-frequency and multidimensional ARKN methods for sys-tems with f depending on both y and y'
2.2.1 Construction of multi-frequency and multidimensional ARKN methods for (2.1)
2.2.2 Stability and phase properties of the novel multi-frequency and multidimensional ARKN methods for oscillatory system (2.1)
2.3 Numerical experiments
2.4 Conclusions
Chapter 3 Extended phase properties and stability analysis of RKN-type integrators for solving general oscillatory second-order initial value problems
3.1 Motivation
3.2 Analysis of dissipation and dispersion through a new test model
3.3 The characteristic matrices of RKN methods and ARKN methods for general oscillatory systems
3.3.1 RKN method
3.3.2 ARKN methods
3.4 Numerical experiments
3.5 Conclusions
Chapter 4 High-order symplectic and symmetric composition methods for multi-frequency and multidimensional oscillatory Hamiltonian systems
4.1 Motivation
4.2 Composition of ARKN integrators for multi-frequency oscillatory sys-tem
4.3 Composition of ERKN integrators
4.4 Numerical experiments
4.5 Conclusions
Chapter 5 An extended discrete gradient formula for oscillatory Hamiltonian systems
5.1 Motivation
5.2 Preliminaries
5.3 An extended discrete gradient formula based on ERKN integrators and its properties
5.4 Convergence of the the fixed-point iteration for the implicit scheme in the discrete gradient formula
5.5 Numerical experiments
5.6 Conclusions
Chapter 6 A linearly-fitted conservative(dissipative)scheme for conservative(dissi-pative)nonlinear wave partial differential equations
6.1 Motivation
6.2 Preliminaries
6.3 Extended discrete gradient method and its remedy
6.4 Numerical experiments
6.4.1 Evaluation of the AVF and choice of starting approximations for fixed-point iteration
6.4.2 Conservative wave equations
6.4.3 Dissipative wave equations
6.5 Conclusions
Bibliography
Foundations and publications
本文编号：3300945
【文章来源】：南京大学江苏省 211工程院校 985工程院校 教育部直属院校
【文章页数】：163 页
【学位级别】：博士
【文章目录】：
Acknowledgements
摘要
Abstract
Chapter 1 Introduction
1.1 Numerical methods for oscillatory problems
1.2 Conservative/dissipative systems and geometric numerical integration
1.3 Layout of thesis
Chapter 2 Multi-frequency and multdimensionalARKN methods for general oscillatorysecond-order initial value problems
2.1 Multi-frequency and multidimensional ARKN methods and the corre-sponding order conditions
2.2 Novel multi-frequency and multidimensional ARKN methods for sys-tems with f depending on both y and y'
2.2.1 Construction of multi-frequency and multidimensional ARKN methods for (2.1)
2.2.2 Stability and phase properties of the novel multi-frequency and multidimensional ARKN methods for oscillatory system (2.1)
2.3 Numerical experiments
2.4 Conclusions
Chapter 3 Extended phase properties and stability analysis of RKN-type integrators for solving general oscillatory second-order initial value problems
3.1 Motivation
3.2 Analysis of dissipation and dispersion through a new test model
3.3 The characteristic matrices of RKN methods and ARKN methods for general oscillatory systems
3.3.1 RKN method
3.3.2 ARKN methods
3.4 Numerical experiments
3.5 Conclusions
Chapter 4 High-order symplectic and symmetric composition methods for multi-frequency and multidimensional oscillatory Hamiltonian systems
4.1 Motivation
4.2 Composition of ARKN integrators for multi-frequency oscillatory sys-tem
4.3 Composition of ERKN integrators
4.4 Numerical experiments
4.5 Conclusions
Chapter 5 An extended discrete gradient formula for oscillatory Hamiltonian systems
5.1 Motivation
5.2 Preliminaries
5.3 An extended discrete gradient formula based on ERKN integrators and its properties
5.4 Convergence of the the fixed-point iteration for the implicit scheme in the discrete gradient formula
5.5 Numerical experiments
5.6 Conclusions
Chapter 6 A linearly-fitted conservative(dissipative)scheme for conservative(dissi-pative)nonlinear wave partial differential equations
6.1 Motivation
6.2 Preliminaries
6.3 Extended discrete gradient method and its remedy
6.4 Numerical experiments
6.4.1 Evaluation of the AVF and choice of starting approximations for fixed-point iteration
6.4.2 Conservative wave equations
6.4.3 Dissipative wave equations
6.5 Conclusions
Bibliography
Foundations and publications
本文编号：3300945
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