Symplectic integrators and their application to dynamical astronomy
Abstract
Symplectic integrators have many merits compared with traditional integrators:
<Heading> </Heading> - 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: <Heading> </Heading> - 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.- Publication:
-
Celestial Mechanics and Dynamical Astronomy
- Pub Date:
- March 1990
- DOI:
- 10.1007/BF00048986
- Bibcode:
- 1990CeMDA..50...59K
- Keywords:
-
- Computational Astrophysics;
- Digital Integrators;
- Many Body Problem;
- Numerical Integration;
- Angular Momentum;
- Differential Equations;
- Runge-Kutta Method;
- Astrophysics;
- Symplectic Integrators;
- Numerical Integration