Structure-Preserving Algorithms for Oscillatory Differential Equations describes a large number of highly effective and efficient structure-preserving algorithms for secondorder oscillatory differential equations by using theoretical analysis and numerical validation.Structure-preserving algorithms for differential equations,especially for oscillatory differential equations,play an important role in the accurate simulation of oscillatory problems in applied sciences and engineering.The book discusses novel advances in the ARKN,ERKN,two-step ERKN,Falkner-type and energy-preserving methods,etc.for oscillatory differential equations.
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目录
1 Runge-Kutta(-Nyström)Methods for Oscillatory Differential Equations 1.1 RK Methods,Rooted Trees,B-Series and Order Conditions 1.2 RKN Methods,Nyström Trees and Order Conditions 1.2.1 Formulation of the Scheme 1.2.2 NyströmTrees andOrderConditions 1.2.3 The Special Case in Absence of the Derivative 1.3 Dispersion and Dissipation of RK(N)Methods 1.3.1 RK Methods 1.3.2 RKN Methods 1.4 Symplectic Methods for Hamiltonian Systems 1.5 Comments on Structure-Preserving Algorithms for Oscillatory Problems References 2 ARKN Methods 2.1 Traditional ARKN Methods 2.1.1 Formulation of the Scheme 2.1.2 OrderConditions 2.2 Symplectic ARKN Methods 2.2.1 SymplecticityConditions forARKNIntegrators 2.2.2 Existence of Symplectic ARKN Integrators 2.2.3 Phase and Stability Properties of Method SARKN1s2 2.2.4 Nonexistence of Symmetric ARKN Methods 2.2.5 Numerical Experiments 2.3 Multidimensional ARKN Methods 2.3.1 Formulation of the Scheme 2.3.2 OrderConditions 2.3.3 Practical Multidimensional ARKN Methods References 3 ERKN Methods 3.1 ERKN Methods 3.1.1 Formulation of Multidimensional ERKN Methods 3.1.2 Special Extended Nyström Tree Theory 3.1.3 OrderConditions 3.2 EFRKN Methods and ERKN Methods 3.2.1 One-Dimensional Case 3.2.2 Multidimensional Case 3.3 ERKN Methods for Second-Order Systems with Variable Principal Frequency Matrix 3.3.1 Analysis Through an Equivalent System 3.3.2 Towards ERKN Methods 3.3.3 Numerical Illustrations References 4 Symplectic and Symmetric Multidimensional ERKN Methods 4.1 Symplecticity and Symmetry Conditions for Multidimensional ERKNIntegrators 4.1.1 SymmetryConditions 4.1.2 SymplecticityConditions 4.2 ConstructionofExplicitSSMERKNIntegrators 4.2.1 Two Two-Stage SSMERKN Integrators of Order Two 4.2.2 AThree-StageSSMERKNIntegratorofOrderFour 4.2.3 Stability and Phase Properties of SSMERKN Integrators 4.3 Numerical Experiments 4.4 ERKN Methods for Long-Term Integration of Orbital Problems 4.5 Symplectic ERKN Methods for Time-Dependent Second-Order Systems 4.5.1 Equivalent Extended Autonomous Systems for Nonautonomous Systems 4.5.2 Symplectic ERKN Methods for Time-Dependent HamiltonianSystems 4.6 Concluding Remarks References 5 Two-Step Multidimensional ERKN Methods 5.1 The Scheifele Two-Step Methods 5.2 Formulation of TSERKN Methods 5.3 OrderConditions 5.3.1 B-Series onSENT 5.3.2 One-StepFormulation 5.3.3 OrderConditions 5.4 Construction of Explicit TSERKN Methods 5.4.1 A Method with Two Function Evaluations per Step 5.4.2 Methods with Three Function Evaluations per Step 5.5 Stability and Phase Properties of the TSERKN Methods 5.6 Numerical Experiments References 6 Adapted Falkner-Type Methods 6.1 Falkner’s Methods 6.2 Formulation of the Adapted Falkner-Type Methods 6.3 ErrorAnalysis 6.4 Stability 6.5 Numerical Experiments Appendix A Derivation of Generating Functions(6.14)and(6.15) Appendix B Proof of(6.24) References 7 Energy-Preserving ERKN Methods 7.1 The Average-Vector-Field Method 7.2 Energy-Preserving ERKN Methods 7.2.1 Formulation of the AAVF methods 7.2.2 A Highly Accurate Energy-Preserving Integrator 7.2.3 Two Properties of the Integrator AAVF-GL 7.3 Numerical Experiment on the Fermi-Pasta-Ulam Problem References 8 Effective Methods for Highly Oscillatory Second-Order Nonlinear Differential Equations 8.1 Numerical Consideration of Highly Oscillatory Second-Order DifferentialEquations 8.2 The Asymptotic Method for Linear Systems 8.3 Waveform Relaxation(WR)Methods for Nonlinear Systems References 9 Extended Leap-Frog Methods for HamiltonianWave Equations 9.1 Conservation Laws and Multi-Symplectic Structures of Wave Equations 9.1.1 Multi-Symplectic Conservation Laws 9.1.2 ConservationLaws forWaveEquations 9.2 ERKNDiscretizationofWaveEquations 9.2.1 Multi-Symplectic Integrators 9.2.2 Multi-Symplectic Extended RKN Discretization 9.3 Explicit Extended Leap-Frog Methods 9.3.1 Eleap-Frog I:An Explicit Multi-Symplectic ERKN Scheme 9.3.2 Eleap-Frog II:An Explicit Multi-Symplectic ERKN-PRK Scheme 9.3.3 Analysis of Linear Stability 9.4 Numerical Experiments 9.4.1 TheConservationLaws andtheSolution 9.4.2 DispersionAnalysis References Appendix First and Second Symposiums on Structure-Preserving Algorithms for Differential Equations,August 2011,June 2012,Nanjing Index
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