Geometric Integrators for Differential Equations with Highly Oscillatory Solutions

The idea of structure-preserving algorithms appeared in the 1980's. The new paradigm brought many innovative changes. The new paradigm wanted to identify the long-time behaviour of the solutions or the existence of conservation laws or some other qualitative feature of the dynamics. Another area that has kept growing in importance within Geometric Numerical Integration is the study of highly-oscillatory problems: problems where the solutions are periodic or quasiperiodic and have to be studied in time intervals that include an extremely large number of periods. As is known, these equations cannot be solved efficiently using conventional methods. A further study of novel geometric integrators has become increasingly important in recent years. The objective of this monograph is to explore further geometric integrators for highly oscillatory problems that can be formulated as systems of ordinary and partial differential equations.

1 Oscillation-Preserving Integrators for Highly Oscillatory Systems of Second-Order ODEs.- 2 Continuous-Stage ERKN Integrators for Second-Order ODEs with Highly Oscillatory Solutions.- 3 Stability and Convergence Analysis of ERKN Integrators for Second-Order ODEs with Highly Oscillatory Solutions.- 4 Functionally-Fitted Energy-Preserving Integrators for Poisson Systems.- 5 Exponential Collocation Methods for Conservative or Dissipative Systems.- 6 Volume-Preserving Exponential Integrators.- 7 Global Error Bounds of One-Stage Explicit ERKN Integrators for Semilinear Wave Equations.- 8 Linearly-Fitted Conservative (Dissipative) Schemes for Nonlinear Wave Equations.- 9 Energy-Preserving Schemes for High-Dimensional Nonlinear KG Equations.- 10 High-Order Symmetric Hermite–Birkhoff Time Integrators for Semilinear KG Equations.- 11 Symplectic Approximations for Efficiently Solving Semilinear KG Equations.- 12 Continuous-Stage Leap-Frog Schemes for Semilinear Hamiltonian Wave Equations.- 13 Semi-Analytical ERKN Integrators for Solving High-Dimensional Nonlinear Wave Equations.- 14 Long-Time Momentum and Actions Behaviour of Energy-Preserving Methods for Wave Equations.