# On the convergence of the spectral viscosity method for the two-dimensional incompressible Euler equations with rough initial data

## TL;DR

This paper introduces a spectral viscosity method for 2D incompressible Euler equations with rough initial data, proving convergence to weak solutions and demonstrating its effectiveness through numerical experiments.

## Contribution

It provides the first convergence proof of a numerical method for Euler equations with rough initial data, bridging theory and computational practice.

## Key findings

- Convergence of the spectral viscosity method to weak solutions with rough initial data.
- Successful numerical simulations of vortex sheets and confined eddies.
- Validation of the method's effectiveness in complex flow scenarios.

## Abstract

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.