| Abstract | 第1-4页 |
| 摘要 | 第4-7页 |
| 1 Introduction | 第7-9页 |
| 2 Preliminaries | 第9-17页 |
| ·Symplectic geometry and symplectic space | 第9-10页 |
| ·Discrization for the Schr¨odinger equations | 第10-13页 |
| ·High order compact method for the spatial approximation | 第10-12页 |
| ·Temporal discrization for the Schr¨odinger equation | 第12-13页 |
| ·Symplectic and Runge-Kutta methods | 第13-17页 |
| ·Symplectic methods | 第13-14页 |
| ·Runge-Kutta methods | 第14-17页 |
| 3 HOC-ADI scheme for multi-dimensional Schr¨odinger equations | 第17-31页 |
| ·HOC-ADI scheme for the 2D LS equations | 第17-21页 |
| ·Construction of HOC-ADI scheme | 第17-18页 |
| ·Theoretical analysis | 第18-21页 |
| ·HOC-ADI splitting approach for 2D NLS equations | 第21-22页 |
| ·Extension to the three-dimensional LS equations | 第22页 |
| ·Numerical examples | 第22-31页 |
| 4 Symplectic Fourier pseudo-spectral schemes for the Klein-Gordon-Schr¨odinger equation | 第31-37页 |
| ·Construction of symplectic Fourier pseudo-spectral scheme | 第31-34页 |
| ·Numerical tests | 第34-37页 |
| 5 HOC splitting multi-symplectic scheme for the CNLS equations | 第37-55页 |
| ·Framework of HOC splitting multi-symplectic scheme | 第37-42页 |
| ·Multi-symplectic structure for the CNLS equations | 第37-39页 |
| ·Splitting multi-symplectic technique for the CNLS equations | 第39-40页 |
| ·Discretization for the linear subproblem | 第40-42页 |
| ·Discretization for the nonlinear subproblem | 第42页 |
| ·HOC-SMS scheme for the CNLS equations | 第42-44页 |
| ·Discrete conservation laws | 第44-46页 |
| ·Numerical Verifications | 第46-55页 |
| 6 Conclusions and Prospects | 第55-57页 |
| Bibliography | 第57-61页 |
| Acknowledgements | 第61-63页 |
| Publications | 第63页 |
