醉染图书高振荡微分方程几何积分法(英文版)9787030671127

ContentsChapter 1 Oscillation-Preserving Integrators For Highly Oscillatory Systems of Second-Order Odes 11.1 Introduction 11.2 Standard Runge-Kutta-Nystrom Schemes From The Matrix-Variation-Of-Constants Formula 51.3 Erkn Integrators And Arkn Methods Based On The Matrix-Variation-Of-Constants Formula 61.3.1 Arkn Integrators 71.3.2 Erkn Integrators 81.4 Oscillation-Preserving Integrators 111.5 Towards Highly Oscillatory Nonlinear Hamiltonian Systems 131.5.1 Ssmerkn Integrators 141.5.2 Trigonometric Fourier Collocation Methods 151.5.3 The Aavf Method And Avf Formula 181.6 Other Concerns Relating To Highly Oscillatory Problems 211.6.1 Gautschi-Iype Methods 211.6.2 General Erkn Methods For (1.1) 211.6.3 Towards The Application To Semilinear Kg Equations 221.7 Numerical Experiments 261.8 Conclusions And Discussion 36References 37Chapter 2 Continuous-Stage Erkn Integrators For Second-Order Odes With Highly Oscillatory Solutions 422.1 Introduction 422.2 Extended Runge-Kutta-Nystrom Methods 45. Continuous-Stage Erkn Methods And Order Conditions 472.4 Energy-Preserving Conditions And Symmetric Conditions 502.5 Linear Stability Analysis 532.6 Construction of Cserkn Methods 552.6.1 The Case of Order Two 562.6.2 The Case of Order Four 572.7 Numerical Experiments 592.8 Conclusions And Discussions 63References 64Chapter 3 Stability And Convergence Analysis of Erkn Integrators For Second-Order Odes With Highly Oscillatory Solutions 683.1 Introduction 683.2 Nonlinear Stability And Convergence Analysis For Erkn Integrators 7.2.1 Nonlinear Stability of The Matrix-Yariation-Of-Constants Formula 7.2.2 Nonlinear Stability And Convergence of Erkn Integrators 773.3 Erkn Integrators With Fourier Pseudospectral Discretisation For Semilinear Wave Equations 833.3.1 Time Discretisation: Erkn Time Integrators 843.3.2 Spatial Discretisation: Fourier Pseudospectral Method 853.3.3 Error Bounds of The Erkn-Fp Method (3.57)-(3.58) 873.4 Numerical Experiments 973.5 Conclusions 107References 107Chapter 4 Functionally-Fitted Energy -Preserving Integrators For Poisson Systems 1114.1 Introduction 1114.2 Functionally-Fitted Ep Integrators 1134.3 Implementation Issues 1154.4 The Existence， Uniqueness And Smoothness 1174.5 Algebraic Order 1204.6 Practical FFEP Integrators 14. Numerical Experiments 1264.8 Conclusions 129References 130Chapter 5 Exponential Collocation Methods For Conservative Or Dissipative Systems 1335.1 Introduction 1335.2 Formulation of Methods 1355.3 Methods For Second-Order Odes With Highly Oscillatory Solutions 1385.4 Energy-Preserving Analysis 1405.5 Existence， Uniqueness And Smoothness of The Solution 1425.6 Algebraic Order 1445.7 Application In Stiff Gradient Systems 1475.8 Practical Examples of Exponential Collocation Methods 1485.8.1 An Example of Ecr Methods 1485.8.2 An Example of Tcr Methods 1485.8.3 An Example of Rkncr Methods 1495.9 Numerical Experiments 1505.10 Concluding Remarks And Discussions 156References 157Chapter 6 Volume-Preserving Exponential Integrators 1616.1 Introduction 1616.2 Exponential Integrators 1636.3 Vp Condition of Exponential Integrators 1646.4 Vp Results For Different Vector Fields 1676.4.1 Vector Fields In 1676.4.2 Vector Fields In 1686.4.3 Vector Fields In 1706.4.4 Vector Fields In (2) 1716.5 Applications To Various Problems 1736.5.1 Highly Oscillatory Second-Order Systems 1736.5.2 Separable Partitioned Systems 1766.5.3 Other Applications 1786.6 Numerical Examples 1796.7 Conclusions 188References 188Chapter 7 Global Error Bounds of One-Stage Explicit Erkn Integrators For Semilinear Wave Equations 1917.1 Introduction 1917.2 Preliminaries 1927.2.1 Spectral Semidiscretisation In Space 1927.2.2 Erkn Integrators 1947.3 Main Result 1957.4 The Lower-Order Error Bounds In Higher-Order Sobolev Spaces 1967.4.1 Regularity Over One Time Step 1967.4.2 Local Error Bound 1977.4.3 Stability 1997.4.4 Proof of Theorem 7.1 Fr-≤α≤0 2007.5 Higher-Order Error Bounds In Lower-Order Sobolev Spaces 2017.6 Numerical Experiments 2047.7 Concluding Remarks 207References 207Chapter 8 Linearly-Fitted Conservative (Dissipative) Schemes For Nonlinear Wave Equations 2108.1 Introduction 2108.2 Preliminaries 2128.3 Extended Discrete Gradient Method 2158.4 Numerical Experiments 2218.4.1 Implementation Issues 2228.4.2 Conservative Wave Equations 2.4.3 Dissipative Wave Equations 0.5 Conclusions 2References Chapter 9 Energy-Preserving Schemes For High-Dimensional Nonlinear Kg Equations 5.1 Introduction 5.2 Formulation of Energy-Preserving Schemes 9.3 Error Analysis 2439.4 Analysis of The Nonlinear Stability 2459.5 Convergence 2489.6 Implementation Issues of Kgdg Scheme 2519.7 Numerical Experiments 2559.7.1 One-Dimensional Problems 2559.7.2 Two-Dimensional Problems 2609.8 Concluding Remarks 262References 263Chapter 10 High-Order Symmetric Hermite-Birkhoff Time Integrators For Semiline

Thesubjectofthisbookisgeometricintegratorsfordifferentialequationswithhighlyoscillatorysolutions,includingoscillation-preservingintegrators,continuous-stageERKNintegrators,nonlinearstabilityandconvergenceanalysisofERKNintegrators,functionally-fittedenergy-preservingintegrators,exponentialcollocationmethods,volume-preservingexponentialintegrators,globalerrorboundsofone-stageERKNintegratorsforsemilinearwaveequations,linearly-fittedconservative/dissipativeintegrators,energy-preservingschemesforKlein?CGordonequations,Hermite?CBirkhofftimeintegratorsforKlein?CGordonequations,symplecticapproximationsforKlein?CGordonequations,continuous-stagemodifiedleap-frogschemeforhigh-dimensionalHamiltonianwaveequations,semi-analyticalexponentialRKNintegrators,long-timemomentumandactionsbehaviourofenergy-preservingmethods.Thenewgeometricintegratorsareappliedtoproblemswithhighlyoscillatorysolutionsfromsciencesandengineering.