This book describes a variety of highly effective and efficient structure preserving algorithms for second order oscillatory differential equations Such systems arise in many branches of science and en8ineering. and the examples in the book include systems trom quantum physics. celestial mechanics and electronics. To accurately simulate the true behavior of such systems, a numerical akorithm must preserve as much as possible their key structural properties: time reversibility, oscillatiun, symplecticity, and energy and momentum conservacion. The book describes novel advances in RKN methods. ERKN methods. Filon-type asymptotic methods. AVF methods. and tfigonometric Fourier collocation methods. The accuracy and efficiency of each of these algorithms are tested via careful numerical simulations. and their structure preserving properties are rigorously established by theoretical analysis The book also give sin sights into the practical implementation of the methods This book is intended for engineers and scientists investigating oscillatory systems. as well as for teachers and students who are interested in structure preserving algorithms for differential equations.
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目录
Contents1 Matrix-Variation-of-Constants Formula 1 1.1 Multi-frequency and Multidimensional Problems1 1.2 Matrix-Variation-of-Constants Formula3 1.3 Towards Classical Runge-Kutta-Nystr?m Schemes8 1.4 Towards ARKN Schemes and ERKN Integrators 9 1.4.1 ARKN Schemes9 1.4.2 ERKN Integrators 10 1.5 Towards Two-Step Multidimensional ERKN Methods11 1.6 Towards AAVF Methods for Multi-frequency OscillatoryHamiltonian Systems 13 1.7 Towards Filon-Type Methods for Multi-frequency HighlyOscillatory Systems14 1.8 Towards ERKN Methods for General Second-OrderOscillatory Systems16 1.9 Towards High-Order Explicit Schemes for HamiltonianNonlinear Wave Equations17 1.10 Conclusions and Discussions 18 References20 2 Improved St?rmer–Verlet Formulae with Applications23 2.1 Motivation23 2.2 Two Improved St?rmer–Verlet Formulae 26 2.2.1 Improved St?rmer–Verlet Formula 126 2.2.2 Improved St?rmer–Verlet Formula 229 2.3 Stability and Phase Properties 31 2.4 Applications 33 2.4.1 Application 1: Time-Independent Schr?dingerEquations 34 2.4.2 Application 2: Non-linear Wave Equations35 2.4.3 Application 3: Orbital Problems 37 2.4.4 Application 4: Fermi–Pasta–Ulam Problem 40 2.5 Coupled Conditions for Explicit Symplecticand Symmetric Multi-frequency ERKN Integratorsfor Multi-frequency Oscillatory Hamiltonian Systems42 2.5.1 Towards Coupled Conditions for ExplicitSymplectic and Symmetric Multi-frequencyERKN Integrators 43 2.5.2 The Analysis of Combined Conditionsfor SSMERKN Integrators for Multi-frequencyand Multidimensional Oscillatory HamiltonianSystems44 2.6 Conclusions and Discussions 48 References49 3 Improved Filon-Type Asymptotic Methods for HighlyOscillatory Differential Equations 53 3.1 Motivation53 3.2 Improved Filon-Type Asymptotic Methods 54 3.2.1 Oscillatory Linear Systems 56 3.2.2 Oscillatory Nonlinear Systems59 3.3 Practical Methods and Numerical Experiments 61 3.4 Conclusions and Discussions 66 References67 4 Efficient Energy-Preserving Integrators for Multi-frequencyOscillatory Hamiltonian Systems69 4.1 Motivation69 4.2 Preliminaries71 4.3 The Derivation of the AAVF Formula 73 4.4 Some Properties of the AAVF Formula77 4.4.1 Stability and Phase Properties 77 4.4.2 Other Properties.79 4.5 Some Integrators Based on AAVF Formula83 4.6 Numerical Experiments 87 4.7 Conclusions 91References92 5 An Extended Discrete Gradient Formula for Multi-frequencyOscillatory Hamiltonian Systems95 5.1 Motivation95 5.2 Preliminaries98 5.3 An Extended Discrete Gradient FormulaBased on ERKN Integrators100 5.4 Convergence of the Fixed-Point Iterationfor the Implicit Scheme 104 5.5 Numerical Experiments 1095.6 Conclusions 114References114 6 Trigonometric Fourier Collocation Methodsfor Multi-frequency Oscillatory Systems117 6.1 Motivation117 6.2 Local Fourier Expansion120 6.3 Formulation of TFC Methods 121 6.3.1 The Calculation of I1;j; I2;j122 6.3.2 Discretization124 6.3.3 The TFC Methods125 6.4 Properties of the TFC Methods128 6.4.1 The Order129 6.4.2 The Order of Energy Preservation and QuadraticInvariant Preservation130 6.4.3 Convergence Analysis of the Iteration 133 6.4.4 Stability and Phase Properties 135 6.5 Numerical Experiments 137 6.6 Conclusions and Discussions 146 References146 7 Error Bounds for Explicit ERKN Integratorsfor Multi-frequency Oscillatory Systems149 7.1 Motivation149 7.2 Preliminaries for Explicit ERKN Integrators 150 7.2.1 Explicit ERKN Integrators and Order Conditions 152 7.2.2 Stability and Phase Properties154 7.3 Preliminary Error Analysis 155 7.3.1 Three Elementary Assumptionsand a Gronwall’s Lemma155 7.3.2 Residuals of ERKN Integrators156 7.4 Error Bounds159 7.5 An Explicit Third Order Integrator with Minimal DispersionError and Dissipation Error166 7.6 Numerical Experiments 169 7.7 Conclusions 173References173 8 Error Analysis of Explicit TSERKN Methods for HighlyOscillatory Systems175 8.1 Motivation175 8.2 The Formulation of the New Method 176 8.3 Error Analysis 183 8.4 Stability and Phase Properties 186 8.5 Numerical Experiments 188 8.6 Conclusions 191 References192 9 Highly Accurate Explicit Symplectic ERKN Methodsfor Multi-frequency Oscillatory Hamiltonian Systems193 9.1 Motivation193 9.2 Preliminaries194 9.3 Explicit Symplectic ERKN Methods of Order Fivewith Some Small Residuals196 9.4 Numerical Experiments 204 9.5 Conclusions and Discussions 208 References208 10 Multidimensional ARKN Methods for GeneralMulti-frequency Oscillatory Second-Order IVPs 211 10.1 Motivation211 10.2 Multidimensional ARKN Methods and the CorrespondingOrder Conditions 212 10.3 ARKN Methods for General Multi-frequencyand Multidimensional Oscillatory Systems214 10.3.1 Construction of Multidimensional ARKN Methods.215 10.3.2 Stability and Phase Properties of MultidimensionalARKN Methods220 10.4 Numerical Experiments 222 10.5 Conclusions and Discussions 225 References227 11 A Simplified Nystr?m-Tree Theory for ERKN IntegratorsSolving Oscillatory Systems229 11.1 Motivation229 11.2 ERKN Methods and Related Issues 231 11.3 Higher Order Derivatives of Vector-Valued Functions233 11.3.1 Taylor Series of Vector-Valued Functions233 11.3.2 Kronecker Inner Product234 11.3.3 The Higher Order Derivatives and KroneckerInner Product235 11.3.4 A Definition Associated with the ElementaryDifferentials 236 11.4 The Set of Simplified Special Extended Nystr?m Trees 238 11.4.1 Tree Set SSENT and Related Mappings238 11.4.2 The Set SSENT and the Set of Classical SN-Trees242 11.4.3 The Set SSENT and the Set SENT 245 11.5 B-series and Order Conditions246 11.5.1 B-series247 11.5.2 Order Conditions 249 11.6 Conclusions and Discussions 251References252 12 General Local Energy-Preserving Integratorsfor Multi-symplectic Hamiltonian PDEs 255 12.1 Motivation255 12.2 Multi-symplectic PDEs and Energy-Preserving ContinuousRunge–Kutta Methods256 12.3 Construction of Local Energy-Preserving Algorithmsfor Hamiltonian PDEs 258 12.3.1 Pseudospectral Spatial Discretization258 12.3.2 Gauss-Legendre Collocation Spatial Discretization264 12.4 Local Energy-Preserving Schemes for Coupled NonlinearSchr?dinger Equations268 12.5 Local Energy-Preserving Schemes for 2D NonlinearSchr?dinger Equations272 12.6 Numerical Experiments for Coupled Nonlinear Schr?dingersEquations 275 12.7 Numerical Experiments for 2D Nonlinear Schr?dingerEquations 284 12.8 Conclusions 289 References290 Conference Photo (Appendix) 293 Index 295