Serge Aubry - Physics Department . Technion

Warning about the problem of reliability of numerical simulations for Long
Time Evolution (Arnol’d diffusion regime)
1-Nonlinear terms are responsible for energy exchanges between the Anderson
modes. At small amplitudes, the system becomes nearly integrable because
nonlinearities are weak. Thus, numerical noise becomes relatively important (with
respect to the nonlinear terms) and could play the essential role for generating
(small) energy diffusion.
-2 A well-known similar problem of « Arnol’d diffusion » was encountered for the
numerical integration of solar planetary systems. The problem was solved by using
secular equations involving only slow variables (orbits parameters).
With such techniques Jacques Laskar could integrate with controlled accuracy
over 5. 10
Serge Aubry, Aubry,
LLB, FRANCE
9 years solar system models using a step size of 200-500 years that is
about only 107 integration steps (see J. Laskar Icarus 196 1-15 (2008)) (while
ordinary techniques would require a step size of only 8 days!) and very short
(reliable) runs.