Computer simulations, exact trajectories, and the gravitational N-body problem

Many physical systems of current interest are chaotic, which means that numerical errors in their simulation are exponentially magnified with the passage of time. This could mean that a numerical solution of a chaotic system is the result of nothing but magnified noise, which calls into question the value of such simulations. Although this fact has been well known for a long time, its impact on the validity of simulations is not well understood. The study of shadowing may provide an answer. A shadow is an exact trajectory of a chaotic map or ordinary differential equation that remains close to an approximate solution for a nontrivial duration of time. If it can be shown that a numerical solution has a shadow, then the validity of the solution is strong, in the sense that it can be viewed as an experimental observation of the shadow, which is an exact solution. We present a discussion of shadowing, including an algorithm to find shadows, using the gravitational N-body problem as an example.

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