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