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A Keplerian-based Hamiltonian Splitting for Gravitational $N$-body Simulations

We developed a Keplerian-based Hamiltonian splitting for solving the gravitational $N$-body problem. This splitting allows us to approximate the solution of a general $N$-body problem by a composition of multiple, independently evolved $2$-body problems. While the Hamiltonian splitting is exact, we show that the composition of independent $2$-body problems results in a non-symplectic non-time-symmetric first-order map. A time-symmetric second-order map is then constructed by composing this basic first-order map with its self-adjoint. The resulting method is precise for each individual $2$-body solution and produces quick and accurate results for near-Keplerian $N$-body systems, like planetary systems or a cluster of stars that orbit a supermassive black hole. The method is also suitable for integration of $N$-body systems with intrinsic hierarchies, like a star cluster with primordial binaries. The superposition of Kepler solutions for each pair of particles makes the method excellently suited for parallel computing; we achieve $\gtrsim 64\%$ efficiency for only $8$ particles per core, but close to perfect scaling for $16384$ particles on a $128$ core distributed-memory computer. We present several implementations in \texttt{Sakura}, one of which is publicly available via the AMUSE framework.