Symplectic Integrators: T + V Revisited and Round-Off Reduced. (arXiv:1812.07007v1 [astro-ph.IM])
<a href="http://arxiv.org/find/astro-ph/1/au:+Chambers_J/0/1/0/all/0/1">John E Chambers</a>
Symplectic integrators separate a problem into parts that can be solved in
isolation, alternately advancing these sub-problems to approximate the
evolution of the complete system. Problems with a single, dominant mass can use
mixed-variable symplectic (MVS) integrators that separate the problem into
Keplerian motion of satellites about the primary, and satellite-satellite
interactions. Here, we examine T+V algorithms where the problem is separated
into kinetic T and potential energy V terms. T+V integrators are typically less
efficient than MVS algorithms. This difference is reduced by using different
step sizes for primary-satellite and satellite-satellite interactions. The T+V
method is improved further using 4th and 6th-order algorithms that include
force gradients and symplectic correctors. We describe three 6th-order
algorithms, containing 2 or 3 force evaluations per step, that are competitive
with MVS in some cases. Round-off errors for T+V integrators can be reduced by
several orders of magnitude, at almost no computational cost, using a simple
modification that keeps track of accumulated changes in the coordinates and
momenta. This makes T+V algorithms desirable for long-term, high-accuracy
calculations.
Symplectic integrators separate a problem into parts that can be solved in
isolation, alternately advancing these sub-problems to approximate the
evolution of the complete system. Problems with a single, dominant mass can use
mixed-variable symplectic (MVS) integrators that separate the problem into
Keplerian motion of satellites about the primary, and satellite-satellite
interactions. Here, we examine T+V algorithms where the problem is separated
into kinetic T and potential energy V terms. T+V integrators are typically less
efficient than MVS algorithms. This difference is reduced by using different
step sizes for primary-satellite and satellite-satellite interactions. The T+V
method is improved further using 4th and 6th-order algorithms that include
force gradients and symplectic correctors. We describe three 6th-order
algorithms, containing 2 or 3 force evaluations per step, that are competitive
with MVS in some cases. Round-off errors for T+V integrators can be reduced by
several orders of magnitude, at almost no computational cost, using a simple
modification that keeps track of accumulated changes in the coordinates and
momenta. This makes T+V algorithms desirable for long-term, high-accuracy
calculations.
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