Time-step selection algorithms: adaptivity, control, and signal processing
Applied Numerical Mathematics - The third international conference on the numerical solutions of volterra and delay equations, May 2004, Tempe, AZ
Constant coefficient linear multistep methods with step density control
Journal of Computational and Applied Mathematics
Adaptivity and computational complexity in the numerical solution of ODEs
Journal of Complexity
On quasi-consistent integration by Nordsieck methods
Journal of Computational and Applied Mathematics
Time-step selection algorithms: Adaptivity, control, and signal processing
Applied Numerical Mathematics
Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation
Journal of Computational and Applied Mathematics
Variable-Stepsize Interpolating Explicit Parallel Peer Methods with Inherent Global Error Control
SIAM Journal on Scientific Computing
| Hi-index | 0.01 |
Adaptive step size control is difficult to combine with geometric numerical integration. As classical step size control is based on "past" information only, time symmetry is destroyed and with it the qualitative properties of the method. In this paper we develop completely explicit, reversible, symmetry-preserving, adaptive step size selection algorithms for geometric numerical integrators such as the Störmer--Verlet method. A new step density controller is proposed and analyzed using backward error analysis and reversible perturbation theory. For integrable reversible systems we show that the resulting adaptive method nearly preserves all action variables and, in particular, the total energy for Hamiltonian systems. It has the same excellent long-term behavior as that obtained when constant steps are used. With variable steps, however, both accuracy and efficiency are greatly improved.