Improving the accuracy of simulated chaotic N-body orbits using smoothness

ABSTRACT Symplectic integrators are a foundation to the study of dynamical N-body phenomena, at scales ranging from planetary to cosmological. These integrators preserve the Poincaré invariants of Hamiltonian dynamics. The N-body Hamiltonian has another, perhaps overlooked, symmetry: it is smooth, or, in other words, it has infinite differentiability class order (DCO) for particle separations greater than 0. Popular symplectic integrators, such as hybrid methods or block adaptive stepping methods do not come from smooth Hamiltonians and it is perhaps unclear whether they should. We investigate the importance of this symmetry by considering hybrid integrators, whose DCO can be tuned easily. Hybrid methods are smooth, except at a finite number of phase space points. We study chaotic planetary orbits in a test considered by Wisdom. We find that increasing smoothness, at negligible extra computational cost in particular tests, improves the Jacobi constant error of the orbits by about 5 orders of magnitude in long-term simulations. The results from this work suggest that smoothness of the N-body Hamiltonian is a property worth preserving in simulations.

On the smoothness of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem

We consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on Cm-smoothness of the free boundary are obtained. In particular, a necessary and sufficient condition for infinite differentiability of the free boundary is given.