Mathematics – Commutative Algebra
Scientific paper
Sep 1999
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999dda....31.0803m&link_type=abstract
American Astronomical Society, DDA meeting #31, #08.03
Mathematics
Commutative Algebra
Scientific paper
Most symplectic integrators used in solar-system dynamics are second-order in the time step tau . Typically, the Hamiltonian is divided into a Keplerian piece HA and a smaller perturbative component HB. We can take advantage of the disparity in relative magnitude of these components to define a second small parameter, call it epsilon = frac {mid HBmid }{mid HAmid } << 1, and use this to obtain a ` partially' \enspace higher-order method. Adopting a Lie series approach, one can, for a given order-N method, examine the tau (N+1) , tau (N+2) , etc. error terms. Each of the 2(k) -2 subterms of the coefficient of the tau (k) error term has an associated factor of epsilon raised to a power ranging from linear to k-1. By including adjustable parameters in each evolution operator exp(tau lbrace *,HArbrace ) or exp(tau lbrace *,HBrbrace ) in the trial method (composed of a combination of these operators) that approximates the true Hamiltonian evolution operator exp(tau lbrace *,HA+HBrbrace right ), one can in principle eliminate specified subterms in specified error terms. For example, a second-order method chosen to eliminate the tau (3) subterms linear in epsilon can, depending on the magnitude of epsilon , produce a quasi-third-order method. In practice this process boils down to generating then solving systems of nonlinear polynomial equations particular to the trial method. A computer algebra program has been developed that automates the generation and solution of the equations that result from requesting a specified method of order N. This task is tedious due to the noncommutative algebra involved in the series expansions and subsequent algebraic manipulations, but computers are well-suited for handling such tedium. Once a method, or set of equivalent methods, has been found, the program then generates and solves a second set of equations for parameter solutions whereby subterms of specified powers in epsilon are eliminated for successive tau (N+1) , tau (N+2) , etc. terms in the overall error expression. The project has, in these initial stages, been at least partially successful. Experiences and results to date will be presented.
Chambers John E.
Murison Marc Allen
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