本书系统阐述了高振荡微分方程几何积分法。内容包括：保振荡积分法分析；连续级ERKN积分法；ERKN积分法的收敛性与稳定性分析；函数拟合能量守恒积分法；指数配置法；保体积指数积分法；一级ERKN积分法解半线性波方程的误差界；线性拟合守恒或耗散积分法；解Klein–Gordon方程的能量守恒积分法；解Klein–Gordon方程的Hermite–Birkhoff积分法；解高维哈密顿波方程的连续级leap-frog格式；半解析指数积分法以及能量守恒积分法的长期行为分析。这些新的几何积分法适用于求解来自于科学与工程领域的高振荡微分方程。

Chapter 1 Oscillation-Preserving Integrators for Highly Oscillatory
Systems of Second-Order ODEs
1.1 Introduction
1.2 Standard Runge-Kutta-Nystr?m 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 Se milinear K G Equations
1.7 Numerical Experiments
1.8 Conclusions and Discussion
References
Chapter 2 Continuous-Stage ERKN Integrators for Second-Order ODEs
with Highly Oscillatory Solutions
2.1 Introduction
2.2 Extended Runge-Kutta-Nystr?m Methods
2.3 Continuous-Stage ERK N 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
Chapter 3 Stability and Convergence Analysis of ERK N Integrators for
Second-Order ODEs with Highly Oscillatory Solutions
3.1 Introduction
3.2 Nonlinear Stability and Convergence Analysis for ERK N
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
Chapter 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
Chapter 5 Exponential Collocation Methods for Conservative or
Dissipative Systems
Chapter 6 Volume-Preserving Exponential Integrators
Chapter 7 Global Error Bounds of One-Stage Explicit ERK N Integrators
for Semilinear Wave Equations
References
Chapter 8 Linearly-Fitted Conservative (Dissipative) Schemes for
Nonlinear Wave Equations
Chapter 9 Energy-Preserving Schemes for High-Dimensional Nonlinear
KG Equations
Chapter 10 High-Order Symmetric Hermite-Birkhoff Time Integrators
for Semilinear KG Equations
Chapter 11 Symplectic Approximations for Efficiently Solving Semilinear
KG Equations
Chapter 12 Continuous-Stage Leap-Frog Schemes for Semilinear
Hamiltonian Wave Equations
Chapter 13 Semi-Analytical ERKN Integrators for Solving High
Dimensional Nonlinear Wave Equations
Chapter 14 Long-Time Momentum and Actions Behaviour of Energy
Preserving Methods for Wave Equations
Index