On Simulation Reliability: A Shadowing-Based Timestep Criterion for Collisionless N-Body Simulations

Wayne Hayes

N-body simulations have become a mainstay in modern astrophysics. They have been used to garner understanding of such varied phenomena as chaos in the solar system, to clumping of matter in the early universe. However, even the earliest practitioners realized that the results of
such simulations may be suspect, because the tiniest differences between two simulations (such as what machine the simulation is run on, or old-fashioned numerical errors) can lead to vastly different simulation results. Over the decades, enormous effort has been put into studying and minimizing such errors, and the consensus
today is that, although the microscopic details of large simulations are almost certainly
incorrect, certain macroscopic measures are valid. However, nobody is quite sure which measures are valid and under precisely what conditions; as such, the fundamental reliability of such simulations has yet to be conclusively
demonstrated. In this talk I will review some past results of simulation reliability
and then introduce the concept of shadowing, which was first applied to N-body systems
by Quinlan & Tremaine in 1992. A shadow of a numerical integration is an exact solution that remains close to the numerical solution for a long time. As such, an integration which has a shadow can be viewed as an observation
of an exact trajectory. Unfortunately, it turns out that the full phase-space integration
of a large N-body system is not shadowable. However, it appears that if one is willing to allow that only some particles have
reliable trajectories, then we can demonstrate that the number of reliable particles decays exponentially with time, and that the decay becomes slower with increasing simulation
accuracy. Unfortunately the decay is extremely rapid for collisional systems, so that all
particles have become unshadowable after just a few crossing times. However, preliminary results for collisionless systems appear to indicate that a large majority of particles can be shadowed for tens or even hundreds of crossing times, using perfectly feasible accuracy criteria that are in
common use today, as long as the softening is not too small.

Presentation (PDF File)

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