Geometric Integrators for Differential Equations with Highly Oscillatory Solutions

• Autore: Wu Xinyuan
• Pubblicazione: Singapore : , : Springer Singapore Pte. Limited, , 2021; ©2021
• Descrizione fisica: 1 online resource (507 pages)
• Soggetto topico: Equacions diferencials; Solucions numèriques
• Soggetto genere / forma: Llibres electrònics
• Altri autori: WangBin
• Nota di contenuto: Intro -- Foreword -- Preface -- Contents -- 1 Oscillation-Preserving Integrators for Highly Oscillatory Systems of Second-Order ODEs -- 1.1 Introduction -- 1.2 Standard Runge-Kutta-Nyström Schemes from the Matrix-Variation-of-Constants Formula -- 1.3 ERKN Integrators and ARKN Methods Based on the Matrix-Variation-of-Constants Formula -- 1.3.1 ARKN Integrators -- 1.3.2 ERKN Integrators -- 1.4 Oscillation-Preserving Integrators -- 1.5 Towards Highly Oscillatory Nonlinear Hamiltonian Systems -- 1.5.1 SSMERKN Integrators -- 1.5.2 Trigonometric Fourier Collocation Methods -- 1.5.3 The AAVF Method and AVF Formula -- 1.6 Other Concerns Relating to Highly Oscillatory Problems -- 1.6.1 Gautschi-Type Methods -- 1.6.2 General ERKN Methods for (1.1) -- 1.6.3 Towards the Application to Semilinear KG Equations -- 1.7 Numerical Experiments -- 1.8 Conclusions and Discussion -- References -- 2 Continuous-Stage ERKN Integrators for Second-Order ODEs with Highly Oscillatory Solutions -- 2.1 Introduction -- 2.2 Extended Runge-Kutta-Nyström Methods -- 2.3 Continuous-Stage ERKN Methods and Order Conditions -- 2.4 Energy-Preserving Conditions and Symmetric Conditions -- 2.5 Linear Stability Analysis -- 2.6 Construction of CSERKN Methods -- 2.6.1 The Case of Order Two -- 2.6.2 The Case of Order Four -- 2.7 Numerical Experiments -- 2.8 Conclusions and Discussions -- References -- 3 Stability and Convergence Analysis of ERKN Integrators for Second-Order ODEs with Highly Oscillatory Solutions -- 3.1 Introduction -- 3.2 Nonlinear Stability and Convergence Analysis for ERKN Integrators -- 3.2.1 Nonlinear Stability of the Matrix-Variation-of-Constants Formula -- 3.2.2 Nonlinear Stability and Convergence of ERKN Integrators -- 3.3 ERKN Integrators with Fourier Pseudospectral Discretisation for Semilinear Wave Equations -- 3.3.1 Time Discretisation: ERKN Time Integrators. ; 3.3.2 Spatial Discretisation: Fourier Pseudospectral Method -- 3.3.3 Error Bounds of the ERKN-FP Method (3.57)-(3.58) -- 3.4 Numerical Experiments -- 3.5 Conclusions -- References -- 4 Functionally-Fitted Energy-Preserving Integrators for Poisson Systems -- 4.1 Introduction -- 4.2 Functionally-Fitted EP Integrators -- 4.3 Implementation Issues -- 4.4 The Existence, Uniqueness and Smoothness -- 4.5 Algebraic Order -- 4.6 Practical FFEP Integrators -- 4.7 Numerical Experiments -- 4.8 Conclusions -- References -- 5 Exponential Collocation Methods for Conservative or Dissipative Systems -- 5.1 Introduction -- 5.2 Formulation of Methods -- 5.3 Methods for Second-Order ODEs with Highly Oscillatory Solutions -- 5.4 Energy-Preserving Analysis -- 5.5 Existence, Uniqueness and Smoothness of the Solution -- 5.6 Algebraic Order -- 5.7 Application in Stiff Gradient Systems -- 5.8 Practical Examples of Exponential Collocation Methods -- 5.8.1 An Example of ECr Methods -- 5.8.2 An Example of TCr Methods -- 5.8.3 An Example of RKNCr Methods -- 5.9 Numerical Experiments -- 5.10 Concluding Remarks and Discussions -- References -- 6 Volume-Preserving Exponential Integrators -- 6.1 Introduction -- 6.2 Exponential Integrators -- 6.3 VP Condition of Exponential Integrators -- 6.4 VP Results for Different Vector Fields -- 6.4.1 Vector Fields in H -- 6.4.2 Vector Fields in S -- 6.4.3 Vector Fields in F(∞) -- 6.4.4 Vector Fields in F(2) -- 6.5 Applications to Various Problems -- 6.5.1 Highly Oscillatory Second-Order Systems -- 6.5.2 Separable Partitioned Systems -- 6.5.3 Other Applications -- 6.6 Numerical Examples -- 6.7 Conclusions -- References -- 7 Global Error Bounds of One-Stage Explicit ERKN Integrators for Semilinear Wave Equations -- 7.1 Introduction -- 7.2 Preliminaries -- 7.2.1 Spectral Semidiscretisation in Space -- 7.2.2 ERKN Integrators -- 7.3 Main Result. ; 7.4 The Lower-Order Error Bounds in Higher-Order Sobolev Spaces -- 7.4.1 Regularity Over One Time Step -- 7.4.2 Local Error Bound -- 7.4.3 Stability -- 7.4.4 Proof of Theorem 7.1 for -1 α 0 -- 7.5 Higher-Order Error Bounds in Lower-Order Sobolev Spaces -- 7.6 Numerical Experiments -- 7.7 Concluding Remarks -- References -- 8 Linearly-Fitted Conservative (Dissipative) Schemes for Nonlinear Wave Equations -- 8.1 Introduction -- 8.2 Preliminaries -- 8.3 Extended Discrete Gradient Method -- 8.4 Numerical Experiments -- 8.4.1 Implementation Issues -- 8.4.2 Conservative Wave Equations -- 8.4.3 Dissipative Wave Equations -- 8.5 Conclusions -- References -- 9 Energy-Preserving Schemes for High-Dimensional Nonlinear KG Equations -- 9.1 Introduction -- 9.2 Formulation of Energy-Preserving Schemes -- 9.3 Error Analysis -- 9.4 Analysis of the Nonlinear Stability -- 9.5 Convergence -- 9.6 Implementation Issues of KGDG Scheme -- 9.7 Numerical Experiments -- 9.7.1 One-Dimensional Problems -- 9.7.2 Two-Dimensional Problems -- 9.8 Concluding Remarks -- References -- 10 High-Order Symmetric Hermite-Birkhoff Time Integrators for Semilinear KG Equations -- 10.1 Introduction -- 10.2 The Symmetric and High-Order Hermite-Birkhoff Time Integration Formula -- 10.2.1 The Operator-Variation-of-Constants Formula -- 10.2.2 The Formulation of the Time Integrators -- 10.3 Stability of the Fully Discrete Scheme -- 10.3.1 Linear Stability Analysis -- 10.3.2 Nonlinear Stability Analysis -- 10.4 Convergence of the Fully Discrete Scheme -- 10.4.1 Consistency -- 10.4.2 Convergence -- 10.5 Spatial Discretisation -- 10.6 Waveform Relaxation and Its Convergence -- 10.7 Numerical Experiments -- 10.8 Conclusions and Discussions -- References -- 11 Symplectic Approximations for Efficiently Solving Semilinear KG Equations -- 11.1 Introduction -- 11.2 Abstract Hamiltonian System of ODEs. ; 11.3 Formulation of the Symplectic Approximation -- 11.3.1 The Time Approximation -- 11.3.2 Symplectic Conditions for the Fully Discrete Scheme -- 11.3.3 Error Analysis of the Extended RKN-Type Approximation -- 11.4 Analysis of the Nonlinear Stability -- 11.5 Convergence -- 11.6 Symplectic Extended RKN-Type Approximation Schemes -- 11.6.1 One-Stage Symplectic Approximation Schemes -- 11.6.2 Two-Stage Symplectic Approximation Schemes -- 11.7 Numerical Experiments -- 11.8 Concluding Remarks -- References -- 12 Continuous-Stage Leap-Frog Schemes for Semilinear Hamiltonian Wave Equations -- 12.1 Introduction -- 12.2 A Continuous-Stage Modified Leap-Frog Scheme -- 12.3 Convergence -- 12.4 Energy-Preserving Continuous-Stage Modified LF Schemes -- 12.5 Symplectic Continuous-Stage Modified LF Scheme -- 12.6 Explicit Continuous-Stage Modified LF Scheme -- 12.7 Numerical Experiments -- 12.8 Conclusions and Discussions -- References -- 13 Semi-Analytical ERKN Integrators for Solving High-Dimensional Nonlinear Wave Equations -- 13.1 Introduction -- 13.2 Preliminaries -- 13.3 Fast Implementation of ERKN Integrators -- 13.4 The Case of Symplectic ERKN Integrators -- 13.5 Analysis of Computational Cost and Memory Usage -- 13.5.1 Computational Cost at Each Time Step -- 13.5.2 Occupied Memory and Maximum Number of Spatial Mesh Grids -- 13.6 Numerical Experiments -- 13.7 Conclusions and Discussions -- References -- 14 Long-Time Momentum and Actions Behaviour of Energy-Preserving Methods for Wave Equations -- 14.1 Introduction -- 14.2 Full Discretisation -- 14.2.1 Spectral Semidiscretisation in Space -- 14.2.2 EP Methods in Time -- 14.3 Main Result and Numerical Experiment -- 14.3.1 Main Result -- 14.3.2 Numerical Experiments -- 14.4 The Proof of the Main Result -- 14.4.1 The Outline of the Proof -- 14.4.2 Modulation Equations -- 14.4.3 Reverse Picard Iteration. ; 14.4.4 Rescaling and Estimation of the Nonlinear Terms -- 14.4.5 Reformulation of the Reverse Picard Iteration -- 14.4.6 Bounds of the Coefficient Functions -- 14.4.7 Defects -- 14.4.8 Remainders -- 14.4.9 Almost Invariants -- 14.4.10 From Short to Long-Time Intervals -- 14.5 Analysis for the AAVF Method with a Quadrature Rule -- 14.6 Conclusions and Discussions -- References -- Index.
• Titolo autorizzato: Geometric Integrators for Differential Equations with Highly Oscillatory Solutions
• ISBN: 981-16-0147-X
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996466398403316
• Lo trovi qui: Univ. di Salerno