$N$-body simulations study the dynamics of $N$ particles under the influence of mutual long-distant forces such as gravity. In practice, $N$-body codes will violate Newton's third law if they use either an approximate Poisson solver or individual timesteps. In this study, we construct a novel $N$-body scheme by combining a fast multipole method (FMM) based Poisson solver and a time integrator using a hierarchical Hamiltonian splitting (HHS) technique. We test our implementation for collision-less systems using several problems in galactic dynamics. As a result of the momentum conserving nature of these two key components, the new $N$-body scheme is also momentum conserving. Moreover, we can fully utilize the $\mathcal O(\textit N)$ complexity of FMM with the integrator. With the restored force symmetry, we can improve both angular momentum conservation and energy conservation substantially. The new scheme will be suitable for many applications in galactic dynamics and structure formation. Our implementation, in the code Taichi, is publicly available at this https URL

中文翻译：

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具有独立时间步长的动量守恒 N 体方案

$N$-body 模拟研究 $N$ 粒子在相互长距离作用力（例如重力）的影响下的动力学。在实践中，如果 $N$-body 代码使用近似泊松求解器或单独的时间步长，它们将违反牛顿第三定律。在这项研究中，我们通过将基于快速多极法 (FMM) 的泊松求解器和使用分层哈密顿分裂 (HHS) 技术的时间积分器相结合，构建了一个新的 $N$-body 方案。我们使用银河动力学中的几个问题来测试我们对无碰撞系统的实现。由于这两个关键组件的动量守恒性质，新的 $N$-body 方案也是动量守恒的。此外，我们可以通过积分器充分利用 FMM 的 $\mathcal O(\textit N)$ 复杂性。随着恢复的力对称性，我们可以大大提高角动量守恒和能量守恒。新方案将适用于星系动力学和结构形成方面的许多应用。我们在代码 Taichi 中的实现可在此 https URL 上公开获得