We 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.

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霜：分层四阶前辛积分器的动量守恒CUDA实现

我们提出了一种新颖的四阶正辛积分器的分层公式及其在 GPU 加速的直接求和 N 体代码霜中的数值实现。由于其分层特性，新积分器特别适用于具有大动态范围的仿真。通过计算除牛顿加速度之外的附加梯度项，可以实现四阶辛积分器中的严格正积分器子步骤。所有的力计算和踢球操作都是同步的，因此积分算法显然是动量守恒的。我们还采用时间步长对称化程序来近似恢复具有自适应单个时间步长的时间可逆性。我们在一系列二进制中演示，霜冻将能量保存到 |ΔE/E| 水平的少体和百万体模拟 ∼ 10-10 而线性和角动量的误差实际上可以忽略不计。对于典型的星团模拟，我们发现霜可以很好地扩展到 $N_\mathrm{GPU}^\mathrm{max}\sim 4\times N/10^5$ 个 GPU，使得直接求和 N 体模拟超过 N = 在具有数百个甚至更多 GPU 的系统上可能有 106 个粒子。由于分层集成的性质，在代码中包含开普勒求解器或具有后牛顿校正的正则化积分器以用于近距离接触和二进制文件是很简单的。在具有数百个甚至更多 GPU 的系统上，可以进行超过 N = 106 个粒子的直接求和 N 体模拟。由于分层集成的性质，在代码中包含开普勒求解器或具有后牛顿校正的正则化积分器以用于近距离接触和二进制文件是很简单的。在具有数百个甚至更多 GPU 的系统上，可以进行超过 N = 106 个粒子的直接求和 N 体模拟。由于分层集成的性质，在代码中包含开普勒求解器或具有后牛顿校正的正则化积分器以用于近距离接触和二进制文件是很简单的。