Numerical orbit integration based on Lie series with use of parallel computing techniques

This article outlines necessary steps to perform numerical orbit integrations based on a Lie series approach. Its implementation requires an efficient evaluation of resulting series coefficients. As an example we treat the classical main problem in satellite orbit calculation (_J2$J_2$ only) and the case of a 4×4$4\times 4$-gravity field. All calculations were performed in very high precision with up to 100 significant digits. In comparison to independent third party computations this approach led to superior results referring to the verifiable constancy of various integrals of motion. To achieve a performance similar to classical numerical integrations in terms of acceptable computing time, at least for non-Keplerian motion problems, we exploited parallel computing capabilities. For our examples, run times were improved by several orders of magnitude, depending on the actual chosen precision level (up to a factor of 50,000 in case of double precision). Here we present the mathematical framework of the proposed orbital integration scheme as well as the work flow for its application in a multi-core, parallel computing environment.