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On the convergence of the spectral viscosity method for the incompressible Euler equations with rough initial data

by S. Lanthaler and S. Mishra

(Report number 2019-17)

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.

Keywords: incompressible Euler spectral viscosity vortex sheet

@Techreport{LM19_821,
  author = {S. Lanthaler and S. Mishra},
  title = {On the convergence of the spectral viscosity method for the incompressible Euler equations with rough initial data},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-17},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-17.pdf },
  year = {2019}
}

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