An adaptive N-body algorithm of optimal order

Picard iteration is normally considered a theoretical tool whose primary utility is to establish the existence and uniqueness of solutions to first-order systems of ordinary differential equations (ODEs). However, in 1996, Parker and Sochacki [Neural, Parallel, Sci. Comput. 4 (1996)] published a practical numerical method for a certain class of ODEs, based upon modified Picard iteration, that generates the Maclaurin series of the solution to arbitrarily high order. The applicable class of ODEs consists of first-order, autonomous systems whose right-hand side functions (generators) are projectively polynomial; that is, they can be written as polynomials in the unknowns. The class is wider than might be expected. The method is ideally suited to the classical N-body problem, which is projectively polynomial. Here, we recast the N-body problem in polynomial form and develop a Picard-based algorithm for its solution. The algorithm is highly accurate, parameter-free, and simultaneously adaptive in time and order. Test cases for both benign and chaotic N-body systems reveal that optimal order is dynamic. That is, in addition to dependency upon N and the desired accuracy, optimal order depends upon the configuration of the bodies at any instant.