Structure-Preserving Algorithms for Oscillatory Differential Equations

Structure-Preserving Algorithms for Oscillatory Differential Equations describes a large number of highly effective and efficient structure-preserving algorithms for secondorder oscillatory differential equations by using theoretical analysis and numerical validation.Structure-preserving algorithms for differential equations,especially for oscillatory differential equations,play an important role in the accurate simulation of oscillatory problems in applied sciences and engineering.The book discusses novel advances in the ARKN,ERKN,two-step ERKN,Falkner-type and energy-preserving methods,etc.for oscillatory differential equations 

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