| Acknowledgements | 第4-9页 |
| 摘要 | 第9-13页 |
| Abstract | 第13-16页 |
| 1 Introduction | 第18-30页 |
| 1.1 Oscillatory second-order systems and the corresponding variation-of-constants formula | 第18-21页 |
| 1.2 Hamiltonian systems and some properties | 第21-22页 |
| 1.3 Symplectic,multi-symplectic,energy-preserving methods | 第22-30页 |
| 1.3.1 Symplectic algorithms for Hamiltonian ODEs | 第22-24页 |
| 1.3.2 Conservation laws,multi-symplectic structures and algorithms for Hamiltonian PDEs | 第24-27页 |
| 1.3.3 Energy-preserving methods for Hamiltonian systems | 第27-30页 |
| 2 Integrators for nonlinear oscillatory systems | 第30-54页 |
| 2.1 Motivation | 第30-32页 |
| 2.2 Successive approximations for system of oscillatory second-order dif-ferential equations | 第32-36页 |
| 2.3 Some explicit Gautschi-type integrators | 第36-37页 |
| 2.4 Error analysis for the local approximate systems | 第37-38页 |
| 2.5 Numerical experiments for successive approximations | 第38-44页 |
| 2.6 An effective approach based on local equivalent system | 第44-50页 |
| 2.7 Conclusions | 第50-54页 |
| 3 Symplectic and symmetric ARKN and ERKN methods | 第54-94页 |
| 3.1 Motivation | 第54-56页 |
| 3.2 ARKN methods and the corresponding order conditions | 第56-57页 |
| 3.3 Symplecticity conditions for ARKN methods and the existence of SARKN integrators | 第57-66页 |
| 3.4 Phase and stability properties of SARKN1s2 method | 第66-68页 |
| 3.5 Symmetry analysis of ARKN methods | 第68-69页 |
| 3.6 Numerical experiments for SARKN methods | 第69-74页 |
| 3.7 ERKN methods and the corresponding order conditions | 第74-75页 |
| 3.8 Symmetry and symplecticity conditions for ERKN methods | 第75-78页 |
| 3.9 Construction of SSERKN methods | 第78-82页 |
| 3.9.1 Two-stage SSERKN methods of order 2 | 第78-80页 |
| 3.9.2 A three-stage SSERKN method of order 4 | 第80-82页 |
| 3.10 Phase and stability properties of the SSERKN methods | 第82-85页 |
| 3.11 Numerical experiments for SSERKN methods | 第85-93页 |
| 3.12 Conclusions and discussions | 第93-94页 |
| 4 A fourth-order numerical schemes for nonlinear Hamiltonian wave equa-tions | 第94-116页 |
| 4.1 Motivation | 第94-96页 |
| 4.2 PDEs to ODEs by finite difference discretizations of spatial derivatives | 第96-101页 |
| 4.3 The stability and convergence of the semidiscretization | 第101-106页 |
| 4.3.1 Discrete energy conservation law | 第101-103页 |
| 4.3.2 The Stability and Convergence | 第103-106页 |
| 4.4 Multidimensional ERKN integrators | 第106-109页 |
| 4.5 Numerical experiments | 第109-115页 |
| 4.6 Conclusions | 第115-116页 |
| 5 Multi-symplectic extended leap-frog methods for Hamiltonian PDEs | 第116-152页 |
| 5.1 Motivation | 第116-118页 |
| 5.2 Conservation laws and multi-symplectic structures of Hamiltonian wave equations | 第118-119页 |
| 5.3 Extended RKN discretization of wave equations | 第119-126页 |
| 5.3.1 Extended RKN methods for ODEs | 第119-120页 |
| 5.3.2 Multi-symplectic extended RKN discretization | 第120-126页 |
| 5.4 Construction of explicit multi-symplectic schemes | 第126-136页 |
| 5.4.1 Eleap-frogⅠ:Multi-symplectic ERKN scheme | 第127-132页 |
| 5.4.2 Eleap-frogⅡ:Multi-symplectic ERKN-PRK scheme | 第132-133页 |
| 5.4.3 Analysis of linear stability | 第133-136页 |
| 5.5 Numerical experiments | 第136-151页 |
| 5.5.1 The conservation laws and the solution | 第136-145页 |
| 5.5.2 Dispersion analysis | 第145-151页 |
| 5.6 Conclusion | 第151-152页 |
| 6 A novel energy-preserving scheme for Hamiltonian wave equations | 第152-176页 |
| 6.1 Motivation | 第152-155页 |
| 6.2 An energy-preserving numerical scheme based on FEM and AVF | 第155-164页 |
| 6.2.1 Spatial semidiscretization:Finite element formulation | 第155-160页 |
| 6.2.2 Time discretization:Average Vector Field method | 第160-164页 |
| 6.3 Illustration by a selected piecewise-linear polynomial basis | 第164-167页 |
| 6.4 Numerical experiments | 第167-172页 |
| 6.5 Conclusions and discussions | 第172-176页 |
| 7 Efficient algorithms for computingΦ_0(Ⅴ)andΦ_1(Ⅴ) | 第176-200页 |
| 7.1 Motivation | 第176-179页 |
| 7.2 ARKN and ERKN integrators for multi-frequency and multidimen-sional oscillatory second-order differential systems | 第179-185页 |
| 7.2.1 Basic definitions of multi-frequency and multidimensiona ARKN and ERKN methods | 第179-180页 |
| 7.2.2 Illuminating experiments | 第180-185页 |
| 7.3 Efficient computation of matrix-valued functions Φ_0(Ⅴ)andΦ_1(Ⅴ) | 第185-189页 |
| 7.4 Proper choices of r,s and N | 第189-194页 |
| 7.5 Stability analysis of ARKN and ERKN methods on the basis of approx-imations to Φ_0(Ⅴ)and Φ_1(Ⅴ) | 第194-198页 |
| 7.6 Conclusions | 第198-200页 |
| Bibliography | 第200-213页 |
| Foundations and publications | 第213-215页 |
