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.

   1 Functionally Fitted Continuous Finite Element Methods for Oscillatory Hamiltonian Systems 1

1.1 Introduction 1

1.2 Functionally-Fitted Continuous Finite Element Methods for Hamiltonian Systems 3

1.3 Interpretation as Continuous-Stage Runge–Kutta Methods and the Analysis on the Algebraic Order 6

1.4 Implementation Issues 17

1.5 Numerical Experiments 19

1.6 Conclusions and Discussions 25

References 26

2 Exponential Average-Vector-Field Integrator for Conservative or Dissipative Systems 29

2.1 Introduction 29

2.2 Discrete Gradient Integrators 31

2.3 Exponential Discrete Gradient Integrators 32

2.4 Symmetry and Convergence of the EAVF Integrator 36

2.5 Problems Suitable for EAVF 38

2.5.1 Highly Oscillatory Nonseparable Hamiltonian Systems 38

2.5.2 Second-Order (Damped) Highly Oscillatory System 39

2.5.3 Semi-discrete Conservative or Dissipative PDEs 42

2.6 Numerical Experiments 44

2.7 Conclusions and Discussions 51

References 52

3 Exponential Fourier Collocation Methods for First-Order Differential Equations 55

3.1 Introduction 55

3.2 Formulation of EFCMs 57

3.2.1 Local Fourier Expansion 57

3.2.2 Discretisation 59

3.2.3 The Exponential Fourier Collocation Methods 61

3.3 Connections with Some Existing Methods 63

3.3.1 Connections with HBVMs and Gauss Methods 63

3.3.2 Connection between EFCMs and Radau IIA Methods 64

3.3.3 Connection between EFCMs and TFCMs 66

3.4 Properties of EFCMs 67

3.4.1 The Hamiltonian Case 67

3.4.2 The Quadratic Invariants 69

3.4.3 Algebraic Order 70

3.4.4 Convergence Condition of the Fixed-Point Iteration 72

3.5 A Practical EFCM and Numerical Experiments 74

3.6 Conclusions and Discussions 82

References 83

4 Symplectic Exponential Runge–Kutta Methods for Solving Nonlinear Hamiltonian Systems 85

4.1 Introduction 85

4.2 Symplectic Conditions for ERK Methods 87

4.3 Symplectic ERK Methods 90

4.4 Numerical Experiments 95

4.5 Conclusions and Discussions 104

References 105

5 High-Order Symplectic and Symmetric Composition Integrators for Multi-frequency Oscillatory Hamiltonian Systems 107

5.1 Introduction 107

5.2 Composition of Multi-frequency ARKN Methods 109

5.3 Composition of ERKN Integrators 119

5.4 Numerical Experiments 125

5.5 Conclusions and Discussions 131

References 132

6 The Construction of Arbitrary Order ERKN Integrators via Group Theory 135

6.1 Introduction 135

6.2 Classical RKN Methods and the RKN Group 136

6.3 ERKN Group and Related Issues 140

6.3.1 Construction of ERKN Group 140

6.3.2 The Relation Between the RKN Group G and the ERKN Group X 144

6.4 A Particular Mapping of G into X 145

6.5 Numerical Experiments 155

6.6 Conclusions and Discussions 162

References 163

7 Trigonometric Collocation Methods for Multi-frequency and Multidimensional Oscillatory Systems 167

7.1 Introduction 167

7.2 Formulation of the Methods 168

7.2.1 The Computation of f e~qecjhTT 170

7.2.2 The Computation of I1;j; I2;j; ~Ici ;j 170

7.2.3 The Scheme of Trigonometric Collocation Methods 173

7.3 Properties of the Methods 176

7.3.1 The Order of Energy Preservation 177

7.3.2 The Order of Quadratic Invariant 178

7.3.3 The Algebraic Order 179

7.3.4 Convergence Analysis of the Iterat