Xinyuan Wu、Xiong You、Bin Wang所著的《振荡微分方程的保结构算法(英文版)(精)》反映了二阶振荡微分方程保结构数值解法研究的最近进展和发展动向，系统阐述了作者及其合作者近五年在常微分方程的ARKN方法、ERKN方法、两步ERKN方法、Falkner型方法、辛方法、对称方法、保能量方法以及偏微分方程多辛方法等方面的重要研究成果。从经典的普适性方法到面向于振荡问题的拟合型方法；从单步法到多步法；从常微分方程的数值解法到偏微分方程的多辛算法。

1 Runge-Kutta (-Nystriim) Methods for Oscillatory Differential Equations

 1.1 RK Methods, Rooted Trees, B-Series and Order Conditions

 1.2 RKN Methods, Nystr6m Trees and Order Conditions

1.2.1 Formulation of the Scheme

1.2.2 Nystr6m Trees and Order Conditions

1.2.3 The Special Case in Absence of the Derivative

 1.3 Dispersion and Dissipation of RK(N) Methods

1.3.1 RKMethods

1.3.2 RKN Methods

 1.4 Symplectic Methods for Hamiltonian Systems

 1.5 Comments on Structure-Preserving Algorithms for Oscillatory Problems

 References

2 ARKN Methods

 2.1 Traditional ARKN Methods

2.1.1 Formulation of the Scheme

2.1.2 Order Conditions

 2.2 Symplectic ARKN Methods

2.2.1 Symplecticity Conditions for ARKN Integrators

2.2.2 Existence of Symplectic ARKN Integrators

2.2.3 Phase and Stability Properties of Method SARKNls2

2.2.4 Nonexistence of Symmetric ARKN Methods

2.2.5 Numerical Experiments

 2.3 Multidimensional ARKN Methods

2.3.1 Formulation of the Scheme

2.3.2 Order Conditions

2.3.3 Practical Multidimensional ARKN Methods

 References

 ERKN Methods

 3.1 ERKN Methods

3.1.1 Formulation of Multidimensional ERKN Methods

3.1.2 Special Extended Nystrrm Tree Theory

3.1.3 Order Conditions

 3.2 EFRKN Methods and ERKN Methods

3.2.1 One-Dimensional Case

3.2.2 Multidimensional Case

 3.3 ERKN Methods for Second-Order Systems with Variable Principal Frequency Matrix

3.3.1 Analysis Through an Equivalent System

3.3.2 Towards ERKN Methods

3.3.3 Numerical Illustrations

 References

4 Symplectic and Symmetric Multidimensional ERKN Methods

 4.1 Symplecticity and Symmetry Conditions for Multidimensional ERKN Integrators

4.1.1 Symmetry Conditions

4.1.2 Symplecticity Conditions

 4.2 Construction of Explicit SSMERKN Integrators

4.2.1 Two Two-Stage SSMERKN Integrators of Order Two

4.2.2 A Three-Stage SSMERKN Integrator of Order Four

4.2.3 Stability and Phase Properties of SSMERKN Integrators

 4.3 Numerical Experiments

 4.4 ERKN Methods for Long-Term Integration of Orbital Problems

 4.5 Symplectic ERKN Methods for Time-Dependent Second-Order Systems

4.5.1 Equivalent Extended Autonomous Systems for Non-autonomous Systems

4.5.2 Symplectic ERKN Methods for Time-Dependent Hamiltonian Systems

 4.6 Concluding Remarks References

 Two-Step Multidimensional ERKN Methods

 5.1 The Scheifele Two-Step Methods

 5.2 Formulation of TSERKN Methods

 5.3 Order Conditions

5.3.1 B-Series on SENT

5.3.2 One-Step Formulation

5.3.3 Order Conditions

 5.4 Construction of Explicit TSERKN Methods

5.4.1 A Method with Two Function Evaluations per Step

5.4.2 Methods with Three Function Evaluations per Step

 5.5 Stability and Phase Properties of the TSERKN Methods

 5.6 Numerical Experiments

 References

6 Adapted Falkner-Type Methods

 6.1 Falkner's Methods

 6.2 Formulation of the Adapted Falkner-Type Methods

 6.3 Error Analysis

 6.4 Stability

 6.5 Numerical Experiments

 Appendix A Derivation of Generating Functions (6.14) and (6.15) . .

 Appendix B Proof of (6.24)

 References

7 Energy-Preserving ERKN Methods

 7.1 The Average-Vector-Field Method

 7.2 Energy-Preserving ERKN Methods

7.2.1 Formulation of the AAVF methods

7.2.2 A Highly Accurate Energy-Preserving Integrator

7.2.3 Two Properties of the Integrator AAVF-GL

 7.3 Numerical Experiment on the Fermi-Pasta-Ulam Problem

 References

8 Effective Methods for Highly Oscillatory Second-Order Nonlinear

 Differential Equations

 8.1 Numerical Consideration of Highly Oscillatory Second-Order

 Differential Equations

 8.2 The Asymptotic Method for Linear Systems

 8.3 Waveform Relaxation (WR) Methods for Nonlinear Systems . .

 References

9 Extended Leap-Frog Methods for Hamiltonian Wave Equations..

 9.1 Conservation Laws and Multi-Symplectic Structures of Wave

 Equations

9.1.1 Multi-Symplectic Conservation Laws

9.1.2 Conservation Laws for Wave Equations

 9.2 ERKN Discretization of Wave Equations

9.2.1 Multi-Symplectic Integrators

9.2.2 Multi-Symplectic Extended RKN Discretization

 9.3 Explicit Extended Leap-Frog Methods

9.3.1 Eleap-Frog I: An Explicit Multi-Symplectic ERKN Scheme

9.3.2 Eleap-Frog II: An Explicit Multi-Symplectic ERKN-PRK Scheme

9.3.3 Analysis of Linear Stability

 9.4 Numerical Experiments

9.4.1 The Conservation Laws and the Solution

9.4.2 Dispersion Analysis

 References

Appendix First and Second Symposiums on Structure-Preserving

 Algorithms for Differential Equations, August 2011, June 2012,

 Nanjing

Index