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J. Chambers (NASA Ames Research Center)
Symplectic integrators are widely used for addressing problems in celestial mechanics. To date, the most popular algorithms are the 2nd-order (leapfrog) and 4th-order methods, which require one and three force evaluations per timestep respectively. Of these, the 4th-order integrator generally gives better performance for a given amount of computational effort, but its performance is limited by the fact that some of its substeps have negative coefficients, i.e. they go backwards compared to the direction of the main integration. This implies that the other (positive) coefficients must be large to compensate. Here I describe a new approach to designing symplectic integrators that appears to have been overlooked to date. This approach makes it possible to design a 4th-order integrator with coefficients that all have magnitudes less than unity, substantially reducing the size of the leading error term compared to the conventional algorithm. I will also describe a new 3rd-order integrator with very small error terms which has no analogue among conventional symplectic algorithms. The performance of the new algorithms will be compared with the standard 2nd and 4th-order methods for several test problems.
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Bulletin of the American Astronomical Society, 34, #3
© 2002. The American Astronomical Society.