Astronomy and Astrophysics – Astronomy
Scientific paper
Sep 1998
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998dda....30.0601f&link_type=abstract
American Astronomical Society, DDA meeting #30, #06.01; Bulletin of the American Astronomical Society, Vol. 30, p.1142
Astronomy and Astrophysics
Astronomy
Scientific paper
Symmetric multistep methods (Lambert and Watson 1976) are powerful numerical integrators for a special second order ordinary differential equation, x'' = f(t, x). They are known to produce only a linear longitude error in orbit integration (Quinlan and Tremaine 1990). The methods are not uniquely determined even if its order and explicitness is specified. Usually the rest freedum is spent to maximize the interval of periodicity, (0, H_0(2)) . Here H_0 is the upper bound of stable stepsize (measured in radian) in integrating oscillatory motions, i.e. the case f(t, x) = - omega (2) x. In order to specify the irreducible symmetric multistep methods more effectively, we introduced a new real function g(theta ) equiv -rho (i theta ) / sigma (i theta ) where rho (z) and sigma (z) are the generating (complex) polynomials of the multistep method. By using the characteristics of g, we discovered a necessary and sufficient condition for the convergent, irreducible, symmetric methods to have a nonzero interval of periodicity. The condition is that g''>0 on all the nonzero double roots of g. Also we proved that H_0(2) is given as the least positive local maximum of g(theta ) in the interval (0,pi ]. Based on these, we found the explicit and implicit symmetric integrators of the order of 6 to 16 having the nearly largest H_0. The order, H_0 of the existing formulas, and H_0 of the newly discovered ones are listed as (6, 0.896, 1.174), (8, 0.718, 1.152), (10, 0.415, 0.859), (12, 0214, 0.811), (14, 0.110, 0.685) in the explicit case, and (8, 1.010, 1.621) in the implicit one. Clearly the regions of stability of the formulas newly discovered are significantly larger than those of the existing ones. These new formulas will be useful in long orbital integrations.
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