Lie integrator

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A Lie integrator is a computer program that solves a system of differential equations by means of numerical integration . The method which is used for numerical integration is the Lie integration and thus gives the program its name. Lie integrators are used, among other things, in celestial mechanics to calculate the course of planets.

Basics

The most important requirement for an effective Lie integrator is to find a recursion with which one can quickly calculate the terms of one of the Lie series required for the integration from just a few given parameters. For this purpose, the system of differential equations to which the method is applied must not be too complicated. Furthermore, the problem to be considered must offer the possibility of determining the required accuracy from the current initial conditions in order to be able to use the flexible step size of the Lie integrator.

Application in celestial mechanics

The first Lie integrator, which was used in celestial mechanics and space travel, was described in 1959 by Wolfgang Gröbner and Ferdinand Cap. It is better suited than the previous integration methods. In 1983 A. Hanslmeier and R. Dvorak developed the method in the programming language Fortran for an N-body problem that can only be solved numerically.

Advantages of Lie integrators

In contrast to many other numerical integrators, Lie integrators can change the step size during runtime. This allows the accuracy to be flexibly adapted to the respective situation (in celestial mechanics: arrangement of planets and the distance to one another). The accuracy does not need to be adjusted to the worst possible situation, but it adapts to the respective requirements. This increases the computing speed of Lie integrators, since they only work as precisely (= slowly) as necessary. This nevertheless enables a very high level of accuracy, since only extremely small rounding errors occur with a sufficiently developed Lie series.

See also

Individual evidence

  1. ^ W. Gröbner, F. Cap: The Three-Body Problem Earth-Moon-Spaceship, Xth International Astronautical Congress London 1959 pp 835-836
  2. ↑ F.Reutter and J. Knapp, Investigations on the numerical treatment of initial value problems of ordinary differential equation systems with the help of LIE series and applications to the calculation of multi-body problems, Springer-Vieweg, ISBN 978-3-663-07472-4 . ( Online )

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