Abstract
ABSTRACTWe present a novel hierarchical formulation of the fourth-order forward symplectic integrator and its numerical implementation in the GPU-accelerated direct-summation N-body code frost. The new integrator is especially suitable for simulations with a large dynamical range due to its hierarchical nature. The strictly positive integrator sub-steps in a fourth-order symplectic integrator are made possible by computing an additional gradient term in addition to the Newtonian accelerations. All force calculations and kick operations are synchronous so the integration algorithm is manifestly momentum-conserving. We also employ a time-step symmetrization procedure to approximately restore the time-reversibility with adaptive individual time-steps. We demonstrate in a series of binary, few-body and million-body simulations that frost conserves energy to a level of |ΔE/E| ∼ 10−10 while errors in linear and angular momentum are practically negligible. For typical star cluster simulations, we find that frost scales well up to $N_\mathrm{GPU}^\mathrm{max}\sim 4\times N/10^5$ GPUs, making direct-summation N-body simulations beyond N = 106 particles possible on systems with several hundred and more GPUs. Due to the nature of hierarchical integration, the inclusion of a Kepler solver or a regularized integrator with post-Newtonian corrections for close encounters and binaries in the code is straightforward.
Highlights
Gravitational direct N-body simulations of collisional star clusters have recently reached the million-body era (e.g. Wang et al 2016)
The solution is to set eTTU = 0 and move the double commutator [U, [T, U]] into the Hamiltonian. This is because [U, [T, U]] can be shown to correspond to a calculable scalar function which only depends on the coordinates of the dynamical system (Takahashi & Imada 1984), so it corresponds to an extra potential term G (Dehnen & Hernandez 2017) in the Hamiltonian defined as
We find that the scaling of the FROST code stalls when the number of GPUs reaches approximately NGmPaUx defined as
Summary
Gravitational direct N-body simulations of collisional star clusters have recently reached the million-body era (e.g. Wang et al 2016). These integrators are widely used in the context of gigayear-long simulations of Solar system bodies Another class of symplectic integrators can be derived using hierarchical Hamiltonian splitting (hereafter HHS). The interaction Hamiltonian USF can be placed on the same hierarchy level as the corresponding slow Hamiltonian HS rendering the dynamics of the fast particles generated by HF independent of the slower hierarchy levels This remarkable decoupling of rapidly evolving dynamical sub-systems enables efficient integration of systems with an extreme dynamical range A rather original and surprisingly rarely used solution to avoid negative integration steps in a fourth-order symplectic integrator is to move appropriate terms from the error Hamiltonian Herr into the surrogate Hamiltonian Hin Eq (1).
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