| 摘要 | 第4-7页 |
| Abstract | 第7-10页 |
| Chapter 1 Introduction | 第14-26页 |
| 1.1 Nonlinear partial differential equation | 第14-17页 |
| 1.2 Energy conservation or dissipation of evolutionary PDEs | 第17-19页 |
| 1.3 Structure-preserving algorithms | 第19-23页 |
| 1.3.1 Energy-preserving algorithms | 第20-23页 |
| 1.3.2 Symmetric algorithms | 第23页 |
| 1.4 Outline of this thesis | 第23-26页 |
| Chapter 2 Energy-preserving algorithms for 1D nonlinear Hamiltonian waveequations | 第26-56页 |
| 2.1 Motivation | 第26-29页 |
| 2.2 Energy-conserving spatial semi discretisation | 第29-42页 |
| 2.2.1 Second-order finite difference for spatial derivative | 第29-35页 |
| 2.2.2 Fourth-order finite difference for spatial derivative | 第35-42页 |
| 2.3 Time integrators: general AVF and general AAVF formula | 第42-49页 |
| 2.3.1 General Average Vector Field method | 第42-44页 |
| 2.3.2 General Adapted Average Vector Field method | 第44-48页 |
| 2.3.3 The convergence of the fixed-point iteration | 第48-49页 |
| 2.4 Numerical experiments | 第49-54页 |
| 2.5 Conclusions and discussions | 第54-56页 |
| Chapter 3 Energy-preserving algorithm for 2D nonlinear Hamiltonian waveequations | 第56-78页 |
| 3.1 Motivation | 第56-58页 |
| 3.2 Energy-preserving spatial discretisation | 第58-68页 |
| 3.2.1 Notations and auxiliary lemmas | 第58-62页 |
| 3.2.2 Fourth-order semidiscretisation, stability and convergence | 第62-67页 |
| 3.2.3 Corresponding Hamiltonian ODEs | 第67-68页 |
| 3.3 Time integrators: Average Vector Field formula | 第68-70页 |
| 3.4 Numerical experiments | 第70-74页 |
| 3.4.1 Test problem: linear wave equation | 第71页 |
| 3.4.2 Simulation of 2D sine-Gordon equations | 第71-74页 |
| 3.5 Conclusions and discussions | 第74-78页 |
| Chapter 4 The operator-variation-of-constants formula | 第78-94页 |
| 4.1 Motivation | 第78-80页 |
| 4.2 Useful properties and boundedness of operators φ_j(A),j∈N | 第80-88页 |
| 4.2.1 Definition of the operator-valued functions | 第81-83页 |
| 4.2.2 Analysis of the boundedness | 第83-86页 |
| 4.2.3 Abstract second-order ODEs and its formal solution | 第86-88页 |
| 4.3 Exact energy-preserving scheme | 第88-91页 |
| 4.4 Explanatory examples | 第91-93页 |
| 4.5 Conclusions and discussions | 第93-94页 |
| Chapter 5 Brikhoff-Hermite time integrators | 第94-130页 |
| 5.1 Motivation | 第94-96页 |
| 5.2 Symmetric and arbitrarily high-order time integrators | 第96-103页 |
| 5.2.1 The operator-variation-of-constants formula | 第96-98页 |
| 5.2.2 Formulation of the Brikhoff Hermite time integrators | 第98-103页 |
| 5.3 Spatial discretisation | 第103-107页 |
| 5.4 Stability of the fully discrete scheme | 第107-113页 |
| 5.4.1 Linear stability analysis | 第109-111页 |
| 5.4.2 Nonlinear stability analysis | 第111-113页 |
| 5.5 Convergence of the fully discrete scheme | 第113-119页 |
| 5.6 Waveform relaxation and its convergence | 第119-120页 |
| 5.7 Numerical experiments | 第120-127页 |
| 5.8 Conclusions and discussions | 第127-130页 |
| Chapter 6 Lagrange collocation-type time-stepping methods | 第130-170页 |
| 6.1 Motivation | 第130-131页 |
| 6.2 Formulation of the Lagrange collocation-type time integrators | 第131-139页 |
| 6.2.1 Construction of the time integrators | 第131-135页 |
| 6.2.2 Local error analysis for the time integrators | 第135-139页 |
| 6.3 Spatial discretisation | 第139-141页 |
| 6.4 Nonlinear stability and convergence analysis | 第141-150页 |
| 6.4.1 Analysis of the nonlinear stability | 第141-144页 |
| 6.4.2 Convergence of the fully discrete scheme | 第144-149页 |
| 6.4.3 The convergence of the fixed-point iteration | 第149-150页 |
| 6.5 The application to 2D Dirichlet/Neumann boundary problems | 第150-155页 |
| 6.5.1 2D problem with Dirichlet boundary conditions | 第151-152页 |
| 6.5.2 2D problem with Neumann boundary conditions | 第152-154页 |
| 6.5.3 Abstract differential formulation and spatial discretisation | 第154-155页 |
| 6.6 Numerical experiments | 第155-166页 |
| 6.6.1 1D problem with periodic boundary conditions | 第157-161页 |
| 6.6.2 Simulation of 2D sine-Gordon equation | 第161-166页 |
| 6.7 Conclusions and discussions | 第166-170页 |
| Bibliography | 第170-184页 |
| Awards, foundations and publications | 第184-188页 |
| Acknowledgments | 第188-189页 |
