Structure-preserving algorithms for oscillatory differential equations

Cover......Page 1
Structure-Preserving Algorithms for Oscillatory Differential Equations......Page 3
Preface......Page 5
Contents......Page 8
1.1 RK Methods, Rooted Trees, B-Series and Order Conditions......Page 12
1.2.1 Formulation of the Scheme......Page 19
1.2.2 Nyström Trees and Order Conditions......Page 20
1.2.3 The Special Case in Absence of the Derivative......Page 29
1.3 Dispersion and Dissipation of RK(N) Methods......Page 30
1.3.1 RK Methods......Page 31
1.3.2 RKN Methods......Page 32
1.4 Symplectic Methods for Hamiltonian Systems......Page 33
1.5 Comments on Structure-Preserving Algorithms for Oscillatory Problems......Page 34
References......Page 35
2.1 Traditional ARKN Methods......Page 37
2.1.1 Formulation of the Scheme......Page 38
2.1.2 Order Conditions......Page 39
2.2 Symplectic ARKN Methods......Page 42
2.2.1 Symplecticity Conditions for ARKN Integrators......Page 43
2.2.2 Existence of Symplectic ARKN Integrators......Page 47
2.2.3 Phase and Stability Properties of Method SARKN1s2......Page 51
2.2.4 Nonexistence of Symmetric ARKN Methods......Page 53
2.2.5 Numerical Experiments......Page 54
2.3.1 Formulation of the Scheme......Page 58
2.3.2 Order Conditions......Page 62
2.3.3 Practical Multidimensional ARKN Methods......Page 67
References......Page 70
3.1 ERKN Methods......Page 72
3.1.1 Formulation of Multidimensional ERKN Methods......Page 73
3.1.2 Special Extended Nyström Tree Theory......Page 74
3.1.3 Order Conditions......Page 82
3.2 EFRKN Methods and ERKN Methods......Page 85
3.2.1 One-Dimensional Case......Page 86
3.2.2 Multidimensional Case......Page 89
3.3.1 Analysis Through an Equivalent System......Page 91
3.3.2 Towards ERKN Methods......Page 92
3.3.3 Numerical Illustrations......Page 95
References......Page 97
4.1 Symplecticity and Symmetry Conditions for Multidimensional ERKN Integrators......Page 100
4.1.1 Symmetry Conditions......Page 101
4.1.2 Symplecticity Conditions......Page 103
4.2.1 Two Two-Stage SSMERKN Integrators of Order Two......Page 108
4.2.2 A Three-Stage SSMERKN Integrator of Order Four......Page 110
4.2.3 Stability and Phase Properties of SSMERKN Integrators......Page 112
4.3 Numerical Experiments......Page 114
4.5 Symplectic ERKN Methods for Time-Dependent Second-Order Systems......Page 120
4.5.1 Equivalent Extended Autonomous Systems for Non-autonomous Systems......Page 121
4.5.2 Symplectic ERKN Methods for Time-Dependent Hamiltonian Systems......Page 122
4.6 Concluding Remarks......Page 124
References......Page 126
5.1 The Scheifele Two-Step Methods......Page 129
5.2 Formulation of TSERKN Methods......Page 132
5.3.1 B-Series on SENT......Page 135
5.3.2 One-Step Formulation......Page 140
5.3.3 Order Conditions......Page 141
5.4.1 A Method with Two Function Evaluations per Step......Page 144
5.4.2 Methods with Three Function Evaluations per Step......Page 146
5.5 Stability and Phase Properties of the TSERKN Methods......Page 150
5.6 Numerical Experiments......Page 153
References......Page 156
6.1 Falkner's Methods......Page 158
6.2 Formulation of the Adapted Falkner-Type Methods......Page 159
6.3 Error Analysis......Page 163
6.4 Stability......Page 168
6.5 Numerical Experiments......Page 173
Appendix A:  Derivation of Generating Functions (6.14) and (6.15)......Page 176
Appendix B:  Proof of (6.24)......Page 177
References......Page 178
7.1 The Average-Vector-Field Method......Page 180
7.2.1 Formulation of the AAVF methods......Page 182
7.2.2 A Highly Accurate Energy-Preserving Integrator......Page 183
7.2.3 Two Properties of the Integrator AAVF-GL......Page 185
7.3 Numerical Experiment on the Fermi-Pasta-Ulam Problem......Page 186
References......Page 190
8.1 Numerical Consideration of Highly Oscillatory Second-Order Differential Equations......Page 192
8.2 The Asymptotic Method for Linear Systems......Page 194
8.3 Waveform Relaxation (WR) Methods for Nonlinear Systems......Page 197
References......Page 202
9.1.1 Multi-Symplectic Conservation Laws......Page 204
9.1.2 Conservation Laws for Wave Equations......Page 206
9.2.2 Multi-Symplectic Extended RKN Discretization......Page 207
9.3.1 Eleap-Frog I: An Explicit Multi-Symplectic ERKN Scheme......Page 215
9.3.3 Analysis of Linear Stability......Page 222
9.4.1 The Conservation Laws and the Solution......Page 224
9.4.2 Dispersion Analysis......Page 231
References......Page 236
Structure-Preserving Algorithms for Oscillatory Differential Equations......Page 238
Appendix: First and Second Symposiums on Structure-Preserving Algorithms for Differential Equations, August 2011, June 2012, Nanjing......Page 239
Index......Page 240