| The numerical integration of Newton's equations of motion for self-gravitating systems, particularly in the context of our Solar System's evolution, remains a paradigm for complex dynamics. We implement Stormer's multistep method in backward difference, summed form and perform arithmetic according to a technique we call ``significance ordered computation.'' We achieve results where the truncation error of our 13th order integrator resides well below machine (double) precision and roundoff error accumulation is random, not systematic. In a previous study, we achieved this ``Brouwer's Law'' in integrations of the 2-D Kepler Problem. Here we show that such error growth can be attained in 3-D Solar System integrations. Our integrations are such that the positions of the major planets are known with an estimated error of no more than 2 degrees after one billion years, a precision unmatched by earlier investigations. Further, we show the outer Solar System is not chaotic, as has previously been reported, but rather computational errors in positions grow no faster than t^{3/2} conforming with existing models for stochastic error growth in an otherwise well-behaved ordinary differential equation system. |
Keywords
solar system, long simulations, Stormer, optimal, chaos
Math Review Classification
Primary 65L06
; Secondary 70F10
Last Updated
October 28, 2004
Length
10
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