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Numerical approximation of statistical solutions of planar, incompressible flows

Seminar for Applied Mathematics (SAM), Eidgenössische Technische Hochschule Zürich (ETHZ), Rämistrasse 101, 8092 Zurich, Switzerland

E-mail Address: [email protected]

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Siddhartha Mishra

Seminar for Applied Mathematics (SAM), Eidgenössische Technische Hochschule Zürich (ETHZ), Rämistrasse 101, 8092 Zurich, Switzerland

E-mail Address: [email protected]

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Christoph Schwab

Seminar for Applied Mathematics (SAM), Eidgenössische Technische Hochschule Zürich (ETHZ), Rämistrasse 101, 8092 Zurich, Switzerland

E-mail Address: [email protected]

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https://doi.org/10.1142/S0218202516500597Cited by:6 (Source: Crossref)

Abstract

We present a finite difference-(multi-level) Monte Carlo algorithm to efficiently compute statistical solutions of the two-dimensional incompressible Navier–Stokes equations (NSE), with periodic boundary conditions and for arbitrary high Reynolds number. We propose a reformulation of statistical solutions in the sense of Foiaş and Prodi in the vorticity-stream function form. The vorticity-stream function formulation of the NSE in two-space dimensions is discretized with a finite difference scheme. We obtain a convergence rate error estimate for this approximation which is explicit in the viscosity parameter $ν$ , under realistic assumptions on the solution regularity. 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 can significantly accelerate the computation of statistical solutions in the sense of Foiaş and Prodi, even for very high Reynolds numbers.

Communicated by F. Brezzi

Keywords:

AMSC: 65N30

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