Numerical Integration with Lie Series
Abstract
The aim of this work is the construction of a fast integration method for differential equations (DE), especially the equations of the motion of celestial bodies. Although a number of integration schemes are available none of them seem to be adequate for treating n-body systems with variable masses, which arise in some cosmogonic problems of the early solar system. As a first step we are now able to present a high-speed numerical integration scheme of the classical n-body system. The basic idea of solving differential equations with Lie-series is due to Grobner (1967) but, unfortunately, he did not elaborate on this method and stopped after some numerically unsatisfactory results. We could simplify the calculation of the Lie-terms and derived finally a recurrence formula for the Lie-terms. Whereas Grobner tried to solve the two-body and three (n-body)problem by two different approaches we solved, at first, in an optimal way the 2-body- problem. Then we were able to derive in a quite similar way the solutions of the 3-body and n-body system. Our integration method for planetary motions has two major advantages: First, it is a relatively fast method (about the factor 3-10 faster in comparison with the n-body program by Schubart-Stumpff, which is commonly used by Astronomers). Second, because larger step lengths can be used, roundoff errors are smaller (e.g. step length 135 d for Jupiter).
- Publication:
-
Astronomy and Astrophysics
- Pub Date:
- March 1984
- Bibcode:
- 1984A&A...132..203H