On Computer Algebra Generation of Symplectic Integrator Methods

Mathematics – Commutative Algebra

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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.

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