Contents
1 Functionally Fitted Continuous Finite Element Methods for Oscillatory Hamiltonian Systems 1
1.1 Introduction 1
1.2 Functionally-Fitted Continuous Finite Element Methods for Hamiltonian Systems 3
1.3 Interpretation as Continuous-Stage Runge–Kutta Methods and the Analysis on the Algebraic Order 6
1.4 Implementation Issues 17
1.5 Numerical Experiments 19
1.6 Conclusions and Discussions 25
References 26
2 Exponential Average-Vector-Field Integrator for Conservative or Dissipative Systems 29
2.1 Introduction 29
2.2 Discrete Gradient Integrators 31
2.3 Exponential Discrete Gradient Integrators 32
2.4 Symmetry and Convergence of the EAVF Integrator 36
2.5 Problems Suitable for EAVF 38
2.5.1 Highly Oscillatory Nonseparable Hamiltonian Systems 38
2.5.2 Second-Order (Damped) Highly Oscillatory System 39
2.5.3 Semi-discrete Conservative or Dissipative PDEs 42
2.6 Numerical Experiments 44
2.7 Conclusions and Discussions 51
References 52
3 Exponential Fourier Collocation Methods for First-Order Differential Equations 55
3.1 Introduction 55
3.2 Formulation of EFCMs 57
3.2.1 Local Fourier Expansion 57
3.2.2 Discretisation 59
3.2.3 The Exponential Fourier Collocation Methods 61
3.3 Connections with Some Existing Methods 63
3.3.1 Connections with HBVMs and Gauss Methods 63
3.3.2 Connection between EFCMs and Radau IIA Methods 64
3.3.3 Connection between EFCMs and TFCMs 66
3.4 Properties of EFCMs 67
3.4.1 The Hamiltonian Case 67
3.4.2 The Quadratic Invariants 69
3.4.3 Algebraic Order 70
3.4.4 Convergence Condition of the Fixed-Point Iteration 72
3.5 A Practical EFCM and Numerical Experiments 74
3.6 Conclusions and Discussions 82
References 83
4 Symplectic Exponential Runge–Kutta Methods for Solving Nonlinear Hamiltonian Systems 85
4.1 Introduction 85
4.2 Symplectic Conditions for ERK Methods 87
4.3 Symplectic ERK Methods 90
4.4 Numerical Experiments 95
4.5 Conclusions and Discussions 104
References 105
5 High-Order Symplectic and Symmetric Composition Integrators for Multi-frequency Oscillatory Hamiltonian Systems 107
5.1 Introduction 107
5.2 Composition of Multi-frequency ARKN Methods 109
5.3 Composition of ERKN Integrators 119
5.4 Numerical Experiments 125
5.5 Conclusions and Discussions 131
References 132
6 The Construction of Arbitrary Order ERKN Integrators via Group Theory 135
6.1 Introduction 135
6.2 Classical RKN Methods and the RKN Group 136
6.3 ERKN Group and Related Issues 140
6.3.1 Construction of ERKN Group 140
6.3.2 The Relation Between the RKN Group G and the ERKN Group X 144
6.4 A Particular Mapping of G into X 145
6.5 Numerical Experiments 155
6.6 Conclusions and Discussions 162
References 163
7 Trigonometric Collocation Methods for Multi-frequency and Multidimensional Oscillatory Systems 167
7.1 Introduction 167
7.2 Formulation of the Methods 168
7.2.1 The Computation of f e~qecjhTT 170
7.2.2 The Computation of I1;j; I2;j; ~Ici ;j 170
7.2.3 The Scheme of Trigonometric Collocation Methods 173
7.3 Properties of the Methods 176
7.3.1 The Order of Energy Preservation 177
7.3.2 The Order of Quadratic Invariant 178
7.3.3 The Algebraic Order 179
7.3.4 Convergence Analysis of the Iteration 180
7.3.5 Stability and Phase Properties 181
7.4 Numerical Experiments 182
7.5 Conclusions and Discussions 191
References 191
8 A Compact Tri-Colored Tree Theory for General ERKN Methods 193
8.1 Introduction 193
8.2 General ERKN Methods 195
8.3 The Failure and the Reduction of the EN-T Theory 196
8.4 The Set of Improved Extended-Nystr?m Trees 199
8.4.1 The IEN-T Set and the Related Mappings 199
8.4.2 The IEN-T Set and the N-T Set 202
8.4.3 The IEN-T Set and the EN-T Set 205
8.4.4 The IEN-T Set and the SSEN-T Set 205
8.5 B-Series for the General ERKN Method 205
8.6 The Order Conditions for the General ERKN Method 208
8.7 The Construction of General ERKN Methods 209
8.7.1 Second-Order General ERKN Methods 209
8.7.2 Third-Order General ERKN Methods 210
8.7.3 Fourth-Order General ERKN Methods 212
8.7.4 An Effective Approach to Constructing the General ERKN Methods 213
8.8 Numerical Experiments 214
8.9 Conclusions and Discussions 218
References 218
9 An Integral Formula Adapted to Different Boundary Conditions for Arbitrarily High-Dimensional Nonlinear Klein–Gordon Equations 221
9.1 Introduction 221
9.2 An Integral Formula for Arbitrarily High-Dimensional Klein–Gordon Equations 224
9.2.1 General Case 224
9.2.2 Homogeneous Case 229
9.2.3 Towards Numerical Simulations 229
9.3 The Consistency of the Boundary Conditions for One-dimensional Klein–Gordon Equations 231
9.3.1 Dirichlet Boundary Conditions 231
9.3.2 Neumann Boundary Conditions 235
9.4 Towards Arbitrarily High-Dimensional Klein–Gordon Equations 237
9.4.1 Dirichlet Boundary Conditions 237
9.4.2 Neumann Boundary Conditions 240
9.4.3 Robin Boundary Condition 243
9.5 Illustrative Examples 243
9.6 Conclusions and Discussions 246
References 249
10 An Energy-Preserving and Symmetric Scheme for Nonlinear Hamiltonian Wave Equations 251
10.1 Introduction 251
10.2 Preliminaries 254
10.3 Operator-Variation-of-Constants Formula for Nonlinear Hamiltonian Wave Equations 255
10.4 Exact Energy-Preserving Scheme for Nonlinear Hamiltonian Wave Equations 257
10.5 Illustrative Examples 263
10.6 Conclusions and Discussions 265
References 266
11 Arbitrarily High-Order Time-Stepping Schemes for Nonlinear Klein–Gordon Equations 269
11.1 Introduction 269
11.2 Abstract Ordinary Differential Equation 272
11.3 Formulation of the Lagrange Collocation-Type Time Integrators 276
11.3.1 Construction of the Time Integrators 277
11.3.2 Error Analysis for the Lagrange Collocation-Type Time-Stepping Integrators 279
11.4 Spatial Discretisation 284
11.5 The Analysis of Nonlinear Stability and Convergence for the Fully Discrete Scheme 286
11.5.1 Analysis of the Nonlinear Stability 286
11.5.2 Convergence of the Fully Discrete Scheme 290
11.5.3 The Convergence of the Fixed-Point Iteration 295
11.6 The Application to Two-dimensional Dirichlet or Neumann Boundary Problems 296
11.6.1 2D Klein–Gordon Equation with Dirichlet Boundary Conditions 297
11.6.2 2D Klein–Gordon Equation with Neumann Boundary Conditions 299
11.6.3 Abstract ODE Formulation and Spatial Discretisation 300
11.7 Numerical Experiments 301
11.7.1 One-dimensional Problem with Periodic Boundary Conditions 304
11.7.2 Simulation of 2D Sine–Gordon Equation 308
11.8 Conclusions and Discussions 313
References 314
12 An Essential Extension of the Finite-Energy Condition for ERKN Integrators Solving Nonlinear Wave Equations 317
12.1 Introduction 317
12.2 Preliminaries 320
12.3 Error Analysis for ERKN Integrators Applied to Nonlinear Wave Equations 324
12.4 Numerical Experiments 333
12.5 Conclusions and Discussions 339
References 340
Index 343