高振荡微分方程几何积分法

这本书分为两部分。第一部分讨论高振荡常微分方程，第二部分讨论具有振荡解的偏微分方程的时间积分。第一部分包括六章，求解高振荡常微分方程，第二部分包括八章，为偏微分方程的几何积分提供了一些新的见解。

Chapter 1 Oscillation-Preserving Integrators For Highly Oscillatory Systems of Second-Order Odes 1
1.1 Introduction 1
1.2 Standard Runge-Kutta-Nystrom Schemes From The Matrix-Variation-Of-Constants Formula 5
1.3 Erkn Integrators And Arkn Methods Based On The Matrix-Variation-Of-Constants Formula 6
1.3.1 Arkn Integrators 7
1.3.2 Erkn Integrators 8
1.4 Oscillation-Preserving Integrators 11
1.5 Towards Highly Oscillatory Nonlinear Hamiltonian Systems 13
1.5.1 Ssmerkn Integrators 14
1.5.2 Trigonometric Fourier Collocation Methods 15
1.5.3 The Aavf Method And Avf Formula 18
1.6 Other Concerns Relating To Highly Oscillatory Problems 21
1.6.1 Gautschi-Iype Methods 21
1.6.2 General Erkn Methods For (1.1) 21
1.6.3 Towards The Application To Semilinear Kg Equations 22
1.7 Numerical Experiments 26
1.8 Conclusions And Discussion 36
References 37
Chapter 2 Continuous-Stage Erkn Integrators For Second-Order Odes With Highly Oscillatory Solutions 42
2.1 Introduction 42
2.2 Extended Runge-Kutta-Nystrom Methods 45
2.3 Continuous-Stage Erkn Methods And Order Conditions 47
2.4 Energy-Preserving Conditions And Symmetric Conditions 50
2.5 Linear Stability Analysis 53
2.6 Construction of Cserkn Methods 55
2.6.1 The Case of Order Two 56
2.6.2 The Case of Order Four 57
2.7 Numerical Experiments 59
2.8 Conclusions And Discussions 63
References 64
Chapter 3 Stability And Convergence Analysis of Erkn Integrators For Second-Order Odes With Highly Oscillatory Solutions 68
3.1 Introduction 68
3.2 Nonlinear Stability And Convergence Analysis For Erkn Integrators 72
3.2.1 Nonlinear Stability of The Matrix-Yariation-Of-Constants Formula 72
3.2.2 Nonlinear Stability And Convergence of Erkn Integrators 77
3.3 Erkn Integrators With Fourier Pseudospectral Discretisation For Semilinear Wave Equations 83
3.3.1 Time Discretisation: Erkn Time Integrators 84
3.3.2 Spatial Discretisation: Fourier Pseudospectral Method 85
3.3.3 Error Bounds of The Erkn-Fp Method (3.57)-(3.58) 87
3.4 Numerical Experiments 97
3.5 Conclusions 107
References 107
Chapter 4 Functionally-Fitted Energy -Preserving Integrators For Poisson Systems 111
4.1 Introduction 111
4.2 Functionally-Fitted Ep Integrators 113
4.3 Implementation Issues 115
4.4 The Existence, Uniqueness And Smoothness 117
4.5 Algebraic Order 120
4.6 Practical FFEP Integrators 123
4.7 Numerical Experiments 126
4.8 Conclusions 129
References 130
Chapter 5 Exponential Collocation Methods For Conservative Or Dissipative Systems 133
5.1 Introduction 133
5.2 Formulation of Methods 135
5.3 Methods For Second-Order Odes With Highly Oscillatory Solutions 138
5.4 Energy-Preserving Analysis 140
5.5 Existence, Uniqueness And Smoothness of The Solution 142
5.6 Algebraic Order 144
5.7 Application In Stiff Gradient Systems 147
5.8 Practical Examples of Exponential Collocation Methods 148
5.8.1 An Example of Ecr Methods 148
5.8.2 An Example of Tcr Methods 148
5.8.3 An Example of Rkncr Methods 149
5.9 Numerical Experiments 150
5.10 Concluding Remarks And Discussions 156
References 157
Chapter 6 Volume-Preserving Exponential Integrators 161
6.1 Introduction 161
6.2 Exponential Integrators 163
6.3 Vp Condition of Exponential Integrators 164
6.4 Vp Results For Different Vector Fields 167
6.4.1 Vector Fields In 167
6.4.2 Vector Fields In 168
6.4.3 Vector Fields In 170
6.4.4 Vector Fields In (2) 171
6.5 Applications To Various Problems 173
6.5.1 Highly Oscillatory Second-Order Systems 173
6.5.2 Separable Partitioned Systems 176
6.5.3 Other Applications 178
6.6 Numerical Examples 179
6.7 Conclusions 188
References 188
Chapter 7 Global Error Bounds of One-Stage Explicit Erkn Integrators For Semilinear Wave Equations 191
7.1 Introduction 191
7.2 Preliminaries 192
7.2.1 Spectral Semidiscretisation In Space 192
7.2.2 Erkn Integrators 194
7.3 Main Result 195
7.4 The Lower-Order Error Bounds In Higher-Order Sobolev Spaces 196
7.4.1 Regularity Over One Time Step 196
7.4.2 Local Error Bound 197
7.4.3 Stability 199
7.4.4 Proof of Theorem 7.1 For-1≤α≤0 200
7.5 Higher-Order Error Bounds In Lower-Order Sobolev Spaces 201
7.6 Numerical Experiments 204
7.7 Concluding Remarks 207
References 207
Chapter 8 Linearly-Fitted Conservative (Dissipative) Schemes For Nonlinear Wave Equations 210
8.1 Introduction 210
8.2 Preliminaries 212
8.3 Extended Discrete Gradient Method 215
8.4 Numerical Experiments 221
8.4.1 Implementation Issues 222
8.4.2 Conservative Wave Equations 223
8.4.3 Dissipative Wave Equations 230
8.5 Conclusions 232
References 233
Chapter 9 Energy-Preserving Schemes For High-Dimensional Nonlinear Kg Equations 235
9.1 Introduction 235
9.2 Formulation of Energy-Preserving Schemes 238
9.3 Error Analysis 243
9.4 Analysis of The Nonlinear Stability 245
9.5 Convergence 248
9.6 Implementation Issues of Kgdg Scheme 251
9.7 Numerical Experiments 255
9.7.1 One-Dimensional Problems 255
9.7.2 Two-Dimensional Problems 260
9.8 Concluding Remarks 262
References 263
Chapter 10 High-Order Symmetric Hermite-Birkhoff Time Integrators For Semilinear Kg Equations 267
10.1 Introduction 267
10.2 The Symmetric And High-Order Hermite-Birkhoff Time Integration Formula 269
10.2.1 The Operator-Variation-Of-Constants Formula 269
10.2.2 The Formulation of The Time Integrators 271
10.3 Stability of The Fully Discrete Scheme 278
10.3.1 Linear Stability Analysis 280
10.3.2 Nonlinear Stability Analysis 282
10.4 Convergence of The Fully Discrete Scheme 284
10.4.1 Consistency 284
10.4.2 Convergence 286
10.5 Spatial Discretisation 292
10.6 Waveform Relaxation And Its Convergence 296
10.7 Numerical Experiments 298
10.8 Conclusions And Discussions 308
References 309
Chapter 11 Symplectic Approximations For Efficiently Solving Semilinear Kg Equations 313
11.1 Introduction 313
11.2 Abstract Hamiltonian System of Odes 316
11.3 Formulation of The Symplectic Approximation 317
11.3.1 The Time Approximation 317
11.3.2 Symplectic Conditions For The Fully Discrete Scheme 319
11.3.3 Error Analysis of The Extended Rkn-Type Approximation 322
11.4 Analysis of The Nonlinear Stability 326
11.5 Convergence 329
11.6 Symplectic Extended Rkn-Type Approximation Schemes 333
11.6.1 One-Stage Symplectic Approximation Schemes 333
11.6.2 Two-Stage Symplectic Approximation Schemes 334
11.7 Numerical Experiments 336
11.8 Concluding Remarks 347
References 348
Chapter 12 Continuous-Stage Leap-Frog Schemes For Semilinear Hamiltonian Wave Equations 352
12.1 Introduction 352
12.2 A Continuous-Stage Modified Leap-Frog Scheme 354
12.3 Convergence 359
12.4 Energy-Preserving Continuous-Stage Modified Lf Schemes 364
12.5 Symplectic Continuous-Stage Modified Lf Scheme 366
12.6 Explicit Continuous-Stage Modified Lf Scheme 368
12.7 Numerical Experiments 371
12.8 Conclusions And Discussions 378
References 378
Chapter 13 Semi-Analytical Erkn Integrators For Solving High-Dimensional Nonlinear Wave Equations 383
13.1 Introduction 383
13.2 Preliminaries 388
13.3 Fast Implementation of Erkn Integrators 390
13.4 The Case of Symplectic Erkn Integrators 393
13.5 Analysis of Computational Cost And Memory Usage 397
13.5.1 Computational Cost At Each Time Step 397
13.5.2 Occupied Memory And Maximum Number of Spatial Mesh Grids 399
13.6 Numerical Experiments 401
13.7 Conclusions And Discussions 410
References 411
Chapter 14 Long-Time Momentum And Actions Behaviour of Energy-
Preserving Methods For Wave Equations 414
14.1 Introduction 414
14.2 Full Discretisation 415
14.2.1 Spectral Semidiscretisation In Space 415
14.2.2 Ep Methods In Time 416
14.3 Main Result And Numerical Experiment 417
14.3.1 Main Result 418
14.3.2 Numerical Experiments 420
14.4 The Proof of The Main Result 424
14.4.1 The Outline of The Proof 424
14.4.2 Modulation Equations 425
14.4.3 Reverse Picard Iteration 429
14.4.4 Rescaling And Estimation of The Nonlinear Terms 430
14.4.5 Reformulation of The Reverse Picard Iteration 431
14.4.6 Bounds of The Coefficient Functions 433
14.4.7 Defects 435
14.4.8 Remainders 438
14.4.9 Almost Invariants 439
14.4.10 From Short To Long-Time Intervals 443
14.5 Analysis For The Aavf Method With A Quadrature Rule 443
14.6 Conclusions And Discussions 444
References 445
Index 448