| 摘要 | 第10-13页 |
| Abstract | 第13-16页 |
| Introduction | 第17-25页 |
| Chapter 1 Preliminaries | 第25-35页 |
| 1.1 Numerical schemes for ODEs | 第25-26页 |
| 1.1.1 Runge-Kutta method | 第25-26页 |
| 1.1.2 Collocation method | 第26页 |
| 1.2 Structure-preserving algorithms for the conservative system | 第26-30页 |
| 1.2.1 Runge-Kutta method | 第27页 |
| 1.2.2 Discrete Gradient method | 第27-30页 |
| 1.3 Structure-preserving algorithms for the infinite dimensional Hamiltoniansystem | 第30-35页 |
| 1.3.1 Multi-symplectic Hamiltonian system | 第32-33页 |
| 1.3.2 Concatenating method | 第33-35页 |
| Chapter 2 Local structure-preserving algorithms for the KdV equation | 第35-77页 |
| 2.1 Concatenating construction of the algorithms | 第37-66页 |
| 2.1.1 Multi-symplectic system of the KdV equation | 第37-39页 |
| 2.1.2 Construction of multi-symplectic algorithms | 第39-48页 |
| 2.1.3 Construction of local energy-preserving algorithms | 第48-57页 |
| 2.1.4 Construction of local momentum-preserving algorithms | 第57-66页 |
| 2.2 Stability analysis | 第66-68页 |
| 2.3 Numerical experiments | 第68-76页 |
| 2.4 Conclusions | 第76-77页 |
| Chapter 3 Construction of the local structure-preserving algorithms forthe general multi-symplectic system | 第77-123页 |
| 3.1 Local structure-preserving algorithms for one-dimensional multi-symplectic PDEs | 第79-87页 |
| 3.1.1 Multi-symplectic algorithms in one-dimension | 第79-83页 |
| 3.1.2 Local energy-preserving algorithms in one-dimension | 第83-85页 |
| 3.1.3 Local momentum-preserving algorithms in one-dimension | 第85-87页 |
| 3.2 Local structure-preserving algorithms for two-dimensional multi-symplectic PDEs | 第87-93页 |
| 3.2.1 Two-dimensional multi-symplectic PDEs | 第88-90页 |
| 3.2.2 Multi-symplectic algorithms in two-dimension | 第90-91页 |
| 3.2.3 Local energy-preserving algorithms in two-dimension | 第91-92页 |
| 3.2.4 Local momentum-preserving algorithms in two-dimension | 第92-93页 |
| 3.3 Local structure-preserving algorithms for the nonlinear Schrodinger e-quation in one-dimension | 第93-101页 |
| 3.4 Local structure-preserving algorithms for the Klein-Gordon-Schrodingerequation in one-dimension | 第101-110页 |
| 3.5 Numerical experiments | 第110-121页 |
| 3.5.1 Numerical simulation of the NLS equation | 第110-116页 |
| 3.5.2 Numerical simulation of the KGS equation | 第116-121页 |
| 3.6 Conclusions | 第121-123页 |
| Chapter 4 Analysis of a conservative high-order compact finite differencescheme for the Klein-Gordon-Schrodinger equation | 第123-147页 |
| 4.1 The fourth-order compact finite difference scheme | 第124-128页 |
| 4.2 The conservation property and the priori estimate | 第128-136页 |
| 4.3 The convergence analysis | 第136-142页 |
| 4.4 Numerical experiments | 第142-146页 |
| 4.4.1 Single solitary wave | 第142-144页 |
| 4.4.2 Solitons collision | 第144-146页 |
| 4.5 Conclusions | 第146-147页 |
| Chapter 5 Numerical analysis of a new conservative scheme for the cou-pled nonlinear Schrodinger equations | 第147-184页 |
| 5.1 A new conservative Fourier pseudospectral scheme | 第150-160页 |
| 5.1.1 Space discretization | 第150-152页 |
| 5.1.2 A Crank-Nicolson/ leap-frog methods for the finite-dimensional canon-ical Hamiltonian system | 第152-160页 |
| 5.2 The existence, uniqueness and stability of the scheme | 第160-167页 |
| 5.3 The convergence of the scheme | 第167-174页 |
| 5.4 Numerical experiments | 第174-179页 |
| 5.4.1 Single solitary wave | 第175-176页 |
| 5.4.2 Solitons collision | 第176-179页 |
| 5.5 Conclusions | 第179-184页 |
| Chapter 6 Analysis of a Fourier pseudospectral conservative scheme forthe Klein-Gordon-Schrodinger equation | 第184-211页 |
| 6.1 Fourier pseudospectral conservative scheme | 第184-189页 |
| 6.1.1 Fourier pseudospectral spatial discretization | 第185-188页 |
| 6.1.2 Symmetric discrete gradient method for the finite-dimensional canon-ical Hamiltonian system | 第188-189页 |
| 6.2 The conservative property and the priori estimate | 第189-195页 |
| 6.3 Analysis of the convergence | 第195-205页 |
| 6.4 Numerical experiments | 第205-210页 |
| 6.4.1 Single solitary wave | 第206-208页 |
| 6.4.2 Solitons collision | 第208-210页 |
| 6.5 Conclusions | 第210-211页 |
| Bibliography | 第211-224页 |
| Publications and Finished Papers | 第224-225页 |
| Acknowledgements | 第225页 |
