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 -- 5.9 Numerical Experiments -- 5.10 Concluding Remarks and Discussions -- References -- 6 Volume-Preserving Exponential Integrators -- 6.1 Introduction -- 6.2 Exponential Integrators -- 6.3 VP Condition of Exponential Integrators -- 6.4 VP Results for Different Vector Fields -- 6.4.1 Vector Fields in H -- 6.4.2 Vector Fields in S -- 6.4.3 Vector Fields in F(∞) -- 6.4.4 Vector Fields in F(2) -- 6.5 Applications to Various Problems -- 6.5.1 Highly Oscillatory Second-Order Systems -- 6.5.2 Separable Partitioned Systems -- 6.5.3 Other Applications -- 6.6 Numerical Examples -- 6.7 Conclusions -- References -- 7 Global Error Bounds of One-Stage Explicit ERKN Integrators for Semilinear Wave Equations -- 7.1 Introduction -- 7.2 Preliminaries -- 7.2.1 Spectral Semidiscretisation in Space -- 7.2.2 ERKN Integrators -- 7.3 Main Result. |