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Numerical approximation of statistical solutions of incompressible flow

by F. Leonardi and S. Mishra and Ch. Schwab

(Report number 2015-27)

Abstract
We present a finite difference-(Multi-level) Monte Carlo algorithm to efficiently compute statistical solutions of the two dimensional Navier-Stokes equations, with periodic boundary conditions and for arbitrarily high Reynolds number. We propose a reformulation of statistical solutions in the vorticity-stream function form. The vorticity-stream function formulation is discretized with a finite difference scheme. We obtain a convergence rate error estimate for this approximation. We also prove convergence and complexity estimates, for the (Multi-level) Monte Carlo finite-difference algorithm to compute statistical solutions. Numerical experiments illustrating the validity of our estimates are presented. They show that the Multi-level Monte Carlo algorithm significantly accelerates the computation of statistical solutions, even for very high Reynolds numbers.

Keywords: Numerical methods, incompressible flows, multi-level Monte Carlo methods, statistical solutions

@Techreport{LMS15_617,
  author = {F. Leonardi and S. Mishra and Ch. Schwab},
  title = {Numerical approximation of statistical solutions of incompressible flow},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-27},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-27.pdf },
  year = {2015}
}

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