周期域中空间曲线上的Biot-Savart积分的快速精确评估
Fast and accurate evaluation of Biot-Savart integrals over spatial curves in periodic domains
AI总结：
本文利用Biot-Savart定律与静电学的类比，将Ewald求和方法应用于周期域涡丝模拟，通过高斯滤波和非均匀快速傅里叶变换实现$O(N \log N)$复杂度的快速精确速度场计算。
摘要：
Biot-Savart定律在电磁学和流体动力学等物理背景中具有重要意义。在流体动力学中，当流体的旋转被限制在一组非常细的涡丝上时，该定律描述了由这些物体的空间排列所诱导的速度场。Biot-Savart定律是用于模拟经典和量子流体流动的涡方法的核心。在处理大量涡元时，朴素方法效率低下，这使得它们不足以模拟湍流涡流。在这里，我们利用Biot-Savart定律与静电学之间的直接类比，将分子动力学模拟中常规使用的Ewald求和方法改编为三维周期域中的涡丝模拟。在此背景下，基本思想是将诱导速度分为(i)由高斯滤波涡量场产生的粗粒化速度，和(ii)考虑涡丝附近近奇异行为的短程修正。前者可以使用非均匀快速傅里叶变换算法精确高效地评估。我们提供了作为方法中参数函数的解析精度估计。我们还讨论了如何在动能估计中正确考虑有限的涡核尺寸。通过数值实验，我们验证了所提出方法的精度和守恒性质。此外，我们证明了该方法在广泛的问题规模N上具有$O(N \log N)$的复杂度，显著优于朴素方法的$O(N^2)$成本。
英文摘要：
The Biot-Savart law is relevant in physical contexts including electromagnetism and fluid dynamics. In the latter case, when the rotation of a fluid is confined to a set of very thin vortex filaments, this law describes the velocity field induced by the spatial arrangement of these objects. The Biot-Savart law is at the core of vortex methods used in the simulation of classical and quantum fluid flows. Naive methods are inefficient when dealing with large numbers of vortex elements, which makes them inadequate for simulating turbulent vortex flows. Here we exploit a direct analogy between the Biot-Savart law and electrostatics to adapt Ewald summation methods, routinely used in molecular dynamics simulations, to vortex filament simulations in three-dimensional periodic domains. In this context, the basic idea is to split the induced velocity onto (i) a coarse-grained velocity generated by a Gaussian-filtered vorticity field, and (ii) a short-range correction accounting for near-singular behaviour near the vortices. The former component can be accurately and efficiently evaluated using the nonuniform fast Fourier transform algorithm. Analytical accuracy estimates are provided as a function of the parameters entering the method. We also discuss how to properly account for the finite vortex core size in kinetic energy estimations. Using numerical experiments, we verify the accuracy and the conservation properties of the proposed approach. Moreover, we demonstrate the $O(N \log N)$ complexity of the method over a wide range of problem sizes $N$, considerably better than the $O(N^2)$ cost of a naive approach.
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