1 Functionally Fitted Continuous Finite Element Methods
for Oscillatory Hamiltonian Systems
1.1 Introduction
1.2 Functionally-Fitted Continuous Finite Element Methods
for Hamiltonian Systems
1.3 Interpretation as Continuous-Stage Runge-Kutta Methods
and the Analysis on the Algebraic Order
1.4 Implementation Issues
1.5 Numerical Experiments
1.6 Conclusions and Discussions
References
2 Exponential Average-Vector-Field Integrator for Conservative
or Dissipative Systems
2.1 Introduction
2.2 Discrete Gradient Integrators
2.3 Exponential Discrete Gradient Integrators
2.4 Symmetry and Convergence of the EAVF Integrator
2.5 Problems Suitable for EAVF
2.5.1 Highly Oscillatory Nonseparable Hamiltonian
Systems
2.5.2 Second-Order (Damped) Highly Oscillatory
System
2.5.3 Semi-discrete Conservative or Dissipative PDEs
2.6 Numerical Experiments
2.7 Conclusions and Discussions
References
3 Exponential Fourier Collocation Methods for First-Order
Differential Equations
3.1 Introduction
3.2 Formulation of EFCMs
3.2.1 Local Fourier Expansion
3.2.2 Discretisation
3.2.3 The Exponential Fourier Collocation Methods
3.3 Connections with Some Existing Methods
3.3.1 Connections with HBVMs and Gauss Methods
3.3.2 Connection between EFCMs and Radau
IIA Methods
3.3.3 Connection between EFCMs and TFCMs
3.4 Properties of EFCMs
3.4.1 The Hamiltonian Case
3.4.2 The Quadratic Invariants
3.4.3 Algebraic Order
3.4.4 Convergence Condition of the Fixed-Point
Iteration
3.5 A Practical EFCM and Numerical Experiments
3.6 Conclusions and Discussions
References
4 Symplectic Exponential Runge-Kutta Methods for Solving
Nonlinear Hamiltonian Systems
4.1 Introduction
4.2 Symplectic Conditions for ERK Methods
4.3 Symplectic ERK Methods
4.4 Numerical Experiments
4.5 Conclusions and Discussions
References
5 High-Order Symplectic and Symmetric Composition Integrators
for Multi-frequency Oscillatory Hamiltonian Systems
5.1 Introduction
5.2 Composition of Multi-frequency ARKN Methods
5.3 Composition of ERKN Integrators
5.4 Numerical Experiments
5.5 Conclusions and Discussions
References
6 The Construction of Arbitrary Order ERKN Integrators
via Group Theory
6.1 Introduction
6.2 Classical RKN Methods and the RKN Group
6.3 ERKN Group and Related Issues
6.3.1 Construction of ERKN Group
6.3.2 The Relation Between the RKN Group
G and the ERKN Group D
6.4 A Particular Mapping of G into D
6.5 Numerical Experiments
6.6 Conclusions and Discussions
References
7 Trigonometric Collocation Methods for Multi-frequency
and Multidimensional Oscillatory Systems
7.1 Introduction
7.2 Formulation of the Methods
7.2.1 The Computation of f
7.2.2 The Computation of I
7.2.3 The Scheme of Trigonometric Collocation Methods.
7.3 Properties of the Methods
7.3.1 The Order of Energy Preservation
7.3.2 The Order of Quadratic Invariant
7.3.3 The Algebraic Order
7.3.4 Convergence Analysis of the Iteration
7.3.5 Stability and Phase Properties
7.4 Numerical Experiments
7.5 Conclusions and Discussions
References
8 A Compact Tri-Colored Tree Theory for General ERKN
Methods
8.1 Introduction
8.2 General ERKN Methods
8.3 The Failure and the Reduction of the EN-T Theory
8.4 The Set of Improved Extended-Nystr6m Trees
8.4.1 The IEN-T Set and the Related Mappings
8.4.2 The IEN-T Set and the N-T Set
8.4.3 The IEN-T Set and the EN-T Set
8.4.4 The IEN-T Set and the SSEN-T Set
8.5 B-Series for the General ERKN Method
8.6 The Order Conditions for the General ERKN Method
8.7 The Construction of General ERKN Methods
8.7.1 Second-Order General ERKN Methods
8.7.2 Third-Order General ERKN Methods

Xinyuan Wu、Bin Wang著的《Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations(精)》介绍：The main theme of this book is recent progress in structure-preserving algorithmsfor solving initial value problems of oscillatory differential equations arising in avariety of research areas, such as astronomy, theoretical physics, electronics, quan-tum mechanics and engineering. It systematically describes the latest advances in thedevelopment of structure-preserving integrators for oscillatory differential equa-tions, such as structure-preserving exponential integrators, functionally fitted energy-preserving integrators, exponential Fourier collocation methods, trigono-metric collocation methods, and symmetric and arbitrarily high-order time-steppingmethods. Most of the material presented here is drawn from the recent literature.Theoretical analysis of the newly developed schemes shows their advantages in thecontext of structure preservation. All the new methods introduced in this book areproven to be highly effective compared with the well-known codes in the scientificliterature. This book also addresses challenging problems at the forefront of modernnumerical analysis and presents a wide range of modern tools and techniques.