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Multi-product splitting and Runge-Kutta-Nyström integrators

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Abstract

The splitting of eh(A+B) into a single product of ehA and ehB results in symplectic integrators when A and B are classical Lie operators. However, at high orders, a single product splitting, with exponentially growing number of operators, is very difficult to derive. This work shows that, if the splitting is generalized to a sum of products, then a simple choice of the basis product reduces the problem to that of extrapolation, with analytically known coefficients and only quadratically growing number of operators. When a multi-product splitting is applied to classical Hamiltonian systems, the resulting algorithm is no longer symplectic but is of the Runge-Kutta-Nyström (RKN) type. Multi-product splitting, in conjunction with a special force-reduction process, explains why at orders p = 4 and 6, RKN integrators only need p − 1 force evaluations.

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  1. Department of Physics, Texas A&M University, College Station, TX, 77843, USA

    Siu A. Chin

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Chin, S.A. Multi-product splitting and Runge-Kutta-Nyström integrators. Celest Mech Dyn Astr 106, 391–406 (2010). https://doi.org/10.1007/s10569-010-9255-9

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