| Acknowlegements | 第4-7页 |
| 摘要 | 第7-9页 |
| Abstract | 第9-10页 |
| 1 Introduction | 第11-21页 |
| 1.1 Structure-preserving algorithms | 第11-14页 |
| 1.1.1 Energy-preserving and -decaying methods | 第12-13页 |
| 1.1.2 Hamiltonian ODEs and symplectic methods | 第13-14页 |
| 1.2 Exponential integrators | 第14-15页 |
| 1.3 Functionally fitted methods | 第15-17页 |
| 1.4 Multi-symplectic PDEs and local energy-preserving methods | 第17-19页 |
| 1.5 Original work associated with this thesis | 第19-20页 |
| 1.6 The outline of this thesis | 第20-21页 |
| 2 Exponential AVF method | 第21-47页 |
| 2.1 Background and motivation | 第21-24页 |
| 2.2 Construction of EAVF for conservative and dissipative systems | 第24-29页 |
| 2.3 Properties of EAVF | 第29-31页 |
| 2.4 Problems suitable for the EAVF | 第31-37页 |
| 2.4.1 Highly oscillatory nonseparable Hamiltonian system | 第31-32页 |
| 2.4.2 Second-order(damped)highly oscillatory system | 第32-35页 |
| 2.4.3 Semi-discrete conservative and dissipative PDEs | 第35-37页 |
| 2.5 Numerical experiments | 第37-46页 |
| 2.6 Conclusions and discussions | 第46-47页 |
| 3 Functionally fitted energy-preserving methods | 第47-78页 |
| 3.1 Background and motivation | 第47-49页 |
| 3.2 Functionally fitted continuous finite element methods | 第49-53页 |
| 3.3 Interpretation as continuous-stage Runge-Kutta methods and order | 第53-65页 |
| 3.4 Implementation issues | 第65-68页 |
| 3.5 Numerical experiments | 第68-75页 |
| 3.6 Conclusions and discussions | 第75-78页 |
| 4 Local energy-preserving methods for multi-symplectic PDEs | 第78-118页 |
| 4.1 Background and motivation | 第78-80页 |
| 4.2 Multi-symplectic PDEs and energy-preserving continuous Runge-Kutta methods | 第80-82页 |
| 4.3 Construction of local energy-preserving algorithms for Hamiltonian PDEs | 第82-94页 |
| 4.3.1 Pseudospectral spatial discretization | 第82-88页 |
| 4.3.2 Gauss-Legendre collocation spatial discretization | 第88-94页 |
| 4.4 Local energy-preserving schemes for coupled nonlinear Schrodinger equations | 第94-98页 |
| 4.5 Local energy-preserving schemes for 2D nonlinear Schrodinger equations | 第98-102页 |
| 4.6 Numerical experiments for coupled nonlinear Schrodingers equations | 第102-112页 |
| 4.7 Numerical experiments for 2D nonlinear Schrodinger equations | 第112-116页 |
| 4.8 Conclusions and discussions | 第116-118页 |
| Bibliography | 第118-127页 |
