The subject of this book is geometric integrators for differential equations with highly oscillatory solutions, including oscillation-preserving integrators, continuous-stage ERKN integrators, nonlinear stability and convergence analysis of ERKN integrators, functionally-fitted energy-preserving integrators, exponential collocation methods, volume-preserving exponential integrators, global error bounds of one-stage ERKN integrators for semilinear wave equations, linearly-fitted conservative/dissipative integrators, energy-preserving schemes for Klein–Gordon equations, Hermite–Birkhoff time integrators for Klein–Gordon equations, symplectic approximations for Klein–Gordon equations, continuous-stage modified leap-frog scheme for high-dimensional Hamiltonian wave equations, semi-analytical exponential RKN integrators,long-time momentum and actions behaviour of energy-preserving methods.The new geometric integrators are applied to problems with highly oscillatory solutions from sciences and engineering.
样章试读
暂时还没有任何用户评论
全部咨询(共0条问答)
暂时还没有任何用户咨询内容
目录
Contents 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
京公网安备 11010102004214号