  We propose a spectral viscosity method to approximate the two-dimensional
Euler equations with rough initial data and prove that the method converges to
a weak solution for a large class of initial data, including when the initial
vorticity is in the so-called Delort class i.e. it is a sum of a signed measure
and an integrable function. This provides the first convergence proof for a
numerical method approximating the Euler equations with such rough initial data
and closes the gap between the available existence theory and rigorous
convergence results for numerical methods. We also present numerical
experiments, including computations of vortex sheets and confined eddies, to
illustrate the proposed method.
