  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.
