  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.
