Abstract
Symplectic integrators have many merits compared with traditional integrators:
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- the numerical solutions have a property of area preserving,
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- 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,
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- 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:
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- the angular momentum vector of the nbody problem is exactly conserved,
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- the numerical solution has a property of time reversibility,
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- the truncation errors, especially the secular error in the angle variables, can easily be estimated by an usual perturbation method,
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- 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 ε−1/n with use of two canonical sets of variables,
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- 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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Kinoshita, H., Yoshida, H. & Nakai, H. Symplectic integrators and their application to dynamical astronomy. Celestial Mech Dyn Astr 50, 59–71 (1990). https://doi.org/10.1007/BF00048986
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DOI: https://doi.org/10.1007/BF00048986