The vortex method is a common numerical and theoretical approach used to implement the motion of an ideal flow, in which the vorticity is approximated by a sum of point vortices, so that the Euler equations read as a system of ordinary differential equations. Such a method is well justified in the full plane, thanks to the explicit representation formulas of Biot and Savart. In an exterior domain, we also replace the impermeable boundary by a collection of point vortices generating the circulation around the obstacle. The density of these point vortices is chosen in order that the flow remains tangent at midpoints between adjacent vortices and that the total vorticity around the obstacle is conserved. In this work, we provide a rigorous justification of this method for any smooth exterior domain, one of the main mathematical difficulties being that the Biot--Savart kernel defines a singular integral operator when restricted to a curve (here, the boundary of the domain). We also introduce an alternative method---the fluid charge method---which, as we argue, is better conditioned and therefore leads to significant numerical improvements.

中文翻译：

---

外域二维理想流动的涡旋法

涡流法是一种常用的数值和理论方法，用于实现理想流动的运动，其中涡量由点涡流的总和来近似，因此欧拉方程读作常微分方程组。由于 Biot 和 Savart 的显式表示公式，这种方法在整个平面上是完全合理的。在外部域中，我们还用一组点涡流代替了不可渗透的边界，这些点涡流在障碍物周围产生了循环。选择这些点涡流的密度是为了使流动在相邻涡流之间的中点处保持切线，并且障碍物周围的总涡量保持不变。在这项工作中，我们为任何光滑的外部域提供了这种方法的严格证明，主要的数学难题之一是 Biot-Savart 核在限制为曲线（此处为域的边界）时定义了奇异积分算子。我们还介绍了一种替代方法——流体电荷法——正如我们所认为的那样，它具有更好的条件，因此导致显着的数值改进。