Symplectic integrators and their application to dynamical astronomy

Details Symplectic integrators and their application to dynamical astronomy Symplectic integrators and their application to dynamical astronomy

Mar 1990

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1990cemda..50...59k&link_type=abstract

Celestial Mechanics and Dynamical Astronomy, Volume 50, Issue 1, pp.59-71

Astronomy and Astrophysics

Astronomy

13

Symplectic Integrators, Numerical Integration

Scientific paper

Symplectic integrators have many merits compared with traditional integrators:

- the numerical solutions have a property of area preserving,
- the discretization error in the energy integral does not have a secular term, which means that the accumulated truncation errors in angle variables increase linearly with the time instead of quadratic growth,
- the symplectic integrators can integrate an orbit with high eccentricity without change of step-size. The symplectic integrators discussed in this paper have the following merits in addition to the previous merits:

- the angular momentum vector of the nbody problem is exactly conserved,
- the numerical solution has a property of time reversibility,
- the truncation errors, especially the secular error in the angle variables, can easily be estimated by an usual perturbation method,
- when a Hamiltonian has a disturbed part with a small parameter c as a factor, the step size of an nth order symplectic integrator can be lengthened by a factor ɛ-1/n with use of two canonical sets of variables,
- the number of evaluation of the force function by the 4th order symplectic integrator is smaller than the classical Runge-Kutta integrator method of the same order. The symplectic integrators are well suited to integrate a Hamiltonian system over a very long time span.

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