| 摘要 | 第7-8页 |
| Abstract | 第8-9页 |
| Introduction | 第10-15页 |
| Chapter 1 Preliminaries | 第15-27页 |
| 1.1 Lagrangian equations and Hamiltonian equations | 第15-16页 |
| 1.2 Symplectic integrator | 第16-19页 |
| 1.2.1 Symplectic Runge-Kutta method | 第17-18页 |
| 1.2.2 Variational integrator | 第18-19页 |
| 1.3 Multisymplectic integrator | 第19-27页 |
| 1.3.1 Multisymplectic Hamiltonian PDEs | 第19-21页 |
| 1.3.2 Multisymplectic Runge-Kutta method | 第21-22页 |
| 1.3.3 Multisymplectic variational integrator | 第22-24页 |
| 1.3.4 Multisymplectic pseudospectral method | 第24-27页 |
| Chapter 2 Symplecticity and multisymplecticity of the Crank-Nicolson scheme for the nonlinear Schrodinger equation | 第27-49页 |
| 2.1 Multisymplectic structure of the NLS equation | 第27-29页 |
| 2.2 Multisymplecticity of the Crank-Nicolson scheme | 第29-36页 |
| 2.2.1 Reconstruction by the concatenating method | 第29-33页 |
| 2.2.2 An equivalent variational integrator | 第33-36页 |
| 2.3 Symplecticity of the Crank-Nicolson scheme | 第36-37页 |
| 2.4 Square conservation and convergence analysis | 第37-43页 |
| 2.4.1 Square conservation | 第39-40页 |
| 2.4.2 Convergence analysis | 第40-43页 |
| 2.5 Numerical experiments | 第43-48页 |
| 2.5.1 One soliton | 第44-47页 |
| 2.5.2 Two solitons | 第47-48页 |
| 2.6 Concluding remarks | 第48-49页 |
| Chapter 3 Numerical dispersion analysis of a multisymplectic scheme for the three dimensional Maxwell's equations | 第49-88页 |
| 3.1 Multisymplectic formulation of Maxwell's equations | 第50-53页 |
| 3.2 Multisymplectic Euler-box scheme for the three dimensional Maswell's equations | 第53-61页 |
| 3.3 Discrete energy conservation law and divergence preservation | 第61-68页 |
| 3.3.1 Discrete energy conservation law | 第61-65页 |
| 3.3.2 Divergence-free preservation | 第65-68页 |
| 3.4 Dispersion and non-dissipation property | 第68-74页 |
| 3.4.1 Dispersion relation of the multisymplectic scheme | 第68-72页 |
| 3.4.2 Comparison with Yee's method | 第72-73页 |
| 3.4.3 Non-dissipation property | 第73-74页 |
| 3.5 Some analyses on the numerical dispersion relation | 第74-86页 |
| 3.6 Concluding remarks | 第86-88页 |
| Chapter 4 Geometric numerical integrators for the Degasperis-Procesi e-quation | 第88-112页 |
| 4.1 Symplectic Fourier pseudospectral integrator for the DP equation | 第89-92页 |
| 4.2 Operator splitting method for the DP equation | 第92-96页 |
| 4.3 Numerical experiments | 第96-110页 |
| 4.4 Concluding remarks | 第110-112页 |
| Bibliography | 第112-119页 |
| Publications and Finished Papers | 第119-120页 |
| Acknowledgements | 第120页 |
