Abstract
Computational efficiency demands discretised, hierarchically organised, and individually adaptive time-step sizes (known as the block-step scheme) for the time integration of N-body models. However, most existing N-body codes adapt individual step sizes in a way that violates time symmetry (and symplecticity), resulting in artificial secular dissipation (and often secular growth of energy errors). Using single-orbit integrations, I investigate various possibilities to reduce or eliminate irreversibility from the time stepping scheme. Significant improvements over the standard approach are possible at little extra effort. However, in order to reduce irreversible step-size changes to negligible amounts, such as suitable for long-term integrations of planetary systems, more computational effort is needed, while exact time reversibility appears elusive for discretised individual step sizes.
Highlights
Astrophysical N-body problems typically have a large range of dynamical time scales with factors of 102−4 between the shortest and longest orbital times
Hardly any of the many computer simulations of stellar dynamics, large-scale structure formation, and galaxy formation would have been possible. This technique still suffers from fundamental limitations in that almost all contemporary implementations violate time symmetry at the basic level, by setting the step size equal to its desired value at the beginning of each step
T and h are not naturally synchronised and to achieve time reversibility some form of synchronisation is necessary. This problem of synchronising h and T exists in the case of a single continuous step size
Summary
Astrophysical N-body problems typically have a large range of dynamical time scales with factors of 102−4 between the shortest and longest orbital times. There are two components to such an individual time-stepping method: a time-step function and a time-stepping scheme. The time-stepping scheme, on the other hand, is a method that adapts the individual particle step sizes to follow these time-step criteria as best as possible. There are two important conditions for such a scheme: (1) it should not hinder computational efficiency and (2) it should be time reversible (and/or support symplectic time integration). There are two important conditions for such a scheme: (1) it should not hinder computational efficiency and (2) it should be time reversible (and/or support symplectic time integration1) This latter condition is important to avoid artificial numerical dissipation (Hairer, Lubich & Wanner 2002).
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