Intro; Preface; Contents; 1 Functionally Fitted Continuous Finite Element Methods for Oscillatory Hamiltonian Systems; 1.1 Introduction; 1.2 Functionally-Fitted Continuous Finite Element Methods …; 1.3 Interpretation as Continuous-Stage Runge-Kutta …; 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 EAVF2.5.1 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 Methods3.3.3 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