# Instituto de Investigación en Matemáticas

## On the choice of time‐transformation functions for adaptive

### Ander Murua (Universidad del País Vasco)

Fecha: 15/12/2011 13:00
Lugar: Sala de Grados de la Facultad de Ciencias
Grupo: Departamento de Matemática Aplicada

Abstract:
It is well known that, when numerically integrating Hamiltonian systems by means of a symplectic integrator, standard variable step‐size implementation destroys the good long‐time behaviour of the numerical solution. However, constant step‐size implementation may be highly inefficient when numerically simulating a trajectory with large variations in the time‐scale (typically, when evolving near singularities of the Hamiltonian function). That difficulty can be overcome by applying a symplectic integrator with constant step‐size to a transformed Hamiltonian system that effectively applies a timetransformation of the form $dt/ d au = s( y)$. This requires an apriori choice of an appropriate function s( y) . In the present work, we obtain bounds of the local discretization errors of B‐series methods (including symplectic Runge‐Kutta methods) and try to use them to choose appropriate timetransformation functions s( y) . We apply the main ideas to construct suitable time‐transformation functions s( y) for N‐body systems.