@misc{Reich1997, author = {Reich, Sebastian}, title = {Preservation of Adiabatic Invariants under Symplectic Discretization}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3210}, number = {SC-97-52}, year = {1997}, abstract = {Symplectic methods, like the Verlet method, are a standard tool for the long term integration of Hamiltonian systems as they arise, for example, in molecular dynamics. One of the reasons for the popularity of symplectic methods is the conservation of energy over very long periods of time up to small fluctuations that scale with the order of the method. In this paper, we discuss a qualitative feature of Hamiltonian systems with separated time scales that is also preserved under symplectic discretization. Specifically, highly oscillatory degrees of freedom often lead to almost preserved quantities (adiabatic invariants). Using recent results from backward error analysis and normal form theory, we show that a symplectic method, like the Verlet method, preserves those adiabatic invariants. We also discuss step-size restrictions necessary to maintain adiabatic invariants in practical computations.}, language = {en} } @misc{BarthLeimkuhlerReich1997, author = {Barth, Eric and Leimkuhler, Benedict and Reich, Sebastian}, title = {A Time-Reversible Variable-Stepsize Integrator for Constrained Dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3220}, number = {SC-97-53}, year = {1997}, abstract = {This article considers the design and implementation of variable-timestep methods for simulating holonomically constrained mechanical systems. Symplectic variable stepsizes are briefly discussed, we then consider time-reparameterization techniques employing a time-reversible (symmetric) integration method to solve the equations of motion. We give several numerical examples, including a simulation of an elastic (inextensible, unshearable) rod undergoing large deformations and collisions with the sides of a bounding box. Numerical experiments indicate that adaptive stepping can significantly smooth the numerical energy and improve the overall efficiency of the simulation.}, language = {en} } @misc{NettesheimReich1997, author = {Nettesheim, Peter and Reich, Sebastian}, title = {Symplectic Multiple-Time-Stepping Integrators for Quantum-Classical Molecular Dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3250}, number = {SC-97-56}, year = {1997}, abstract = {The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK.}, language = {en} }