| Acknowledgements | 第4-9页 |
| 摘要 | 第9-11页 |
| Abstract | 第11-12页 |
| Chapter 1 Introduction | 第13-23页 |
| 1.1 Numerical methods for oscillatory problems | 第13-16页 |
| 1.2 Conservative/dissipative systems and geometric numerical integration | 第16-19页 |
| 1.3 Layout of thesis | 第19-23页 |
| Chapter 2 Multi-frequency and multdimensionalARKN methods for general oscillatorysecond-order initial value problems | 第23-43页 |
| 2.1 Multi-frequency and multidimensional ARKN methods and the corre-sponding order conditions | 第24-28页 |
| 2.2 Novel multi-frequency and multidimensional ARKN methods for sys-tems with f depending on both y and y' | 第28-36页 |
| 2.2.1 Construction of multi-frequency and multidimensional ARKN methods for (2.1) | 第28-34页 |
| 2.2.2 Stability and phase properties of the novel multi-frequency and multidimensional ARKN methods for oscillatory system (2.1) | 第34-36页 |
| 2.3 Numerical experiments | 第36-40页 |
| 2.4 Conclusions | 第40-43页 |
| Chapter 3 Extended phase properties and stability analysis of RKN-type integrators for solving general oscillatory second-order initial value problems | 第43-65页 |
| 3.1 Motivation | 第43-45页 |
| 3.2 Analysis of dissipation and dispersion through a new test model | 第45-50页 |
| 3.3 The characteristic matrices of RKN methods and ARKN methods for general oscillatory systems | 第50-56页 |
| 3.3.1 RKN method | 第50-54页 |
| 3.3.2 ARKN methods | 第54-56页 |
| 3.4 Numerical experiments | 第56-62页 |
| 3.5 Conclusions | 第62-65页 |
| Chapter 4 High-order symplectic and symmetric composition methods for multi-frequency and multidimensional oscillatory Hamiltonian systems | 第65-95页 |
| 4.1 Motivation | 第66-68页 |
| 4.2 Composition of ARKN integrators for multi-frequency oscillatory sys-tem | 第68-78页 |
| 4.3 Composition of ERKN integrators | 第78-86页 |
| 4.4 Numerical experiments | 第86-93页 |
| 4.5 Conclusions | 第93-95页 |
| Chapter 5 An extended discrete gradient formula for oscillatory Hamiltonian systems | 第95-123页 |
| 5.1 Motivation | 第95-97页 |
| 5.2 Preliminaries | 第97-101页 |
| 5.3 An extended discrete gradient formula based on ERKN integrators and its properties | 第101-106页 |
| 5.4 Convergence of the the fixed-point iteration for the implicit scheme in the discrete gradient formula | 第106-112页 |
| 5.5 Numerical experiments | 第112-121页 |
| 5.6 Conclusions | 第121-123页 |
| Chapter 6 A linearly-fitted conservative(dissipative)scheme for conservative(dissi-pative)nonlinear wave partial differential equations | 第123-149页 |
| 6.1 Motivation | 第123-124页 |
| 6.2 Preliminaries | 第124-129页 |
| 6.3 Extended discrete gradient method and its remedy | 第129-135页 |
| 6.4 Numerical experiments | 第135-146页 |
| 6.4.1 Evaluation of the AVF and choice of starting approximations for fixed-point iteration | 第135-137页 |
| 6.4.2 Conservative wave equations | 第137-143页 |
| 6.4.3 Dissipative wave equations | 第143-146页 |
| 6.5 Conclusions | 第146-149页 |
| Bibliography | 第149-161页 |
| Foundations and publications | 第161-163页 |
