Barth, Andrea; Schwab, Christoph; Šukys, Jonas Multilevel Monte Carlo simulation of statistical solutions to the Navier-Stokes equations. (English) Zbl 1382.76209 Cools, Ronald (ed.) et al., Monte Carlo and quasi-Monte Carlo methods. MCQMC. Proceedings of the 11th international conference on ‘Monte Carlo and quasi-Monte Carlo methods in scientific computing’, Leuven, Belgium, April 6–11, 2014. Cham: Springer (ISBN 978-3-319-33505-6/hbk; 978-3-319-33507-0/ebook). Springer Proceedings in Mathematics & Statistics 163, 209-227 (2016). Summary: We propose Monte Carlo (MC), single level Monte Carlo (SLMC) and multilevel Monte Carlo (MLMC) methods for the numerical approximation of statistical solutions to the viscous, incompressible Navier-Stokes equations (NSE) on a bounded, connected domain \(D\subset \mathbb {R}^d\), \(d=1,2\) with no-slip or periodic boundary conditions on the boundary \(\partial D\). The MC convergence rate of order 1/2 is shown to hold independently of the Reynolds number with constant depending only on the mean kinetic energy of the initial velocity ensemble. We discuss the effect of space-time discretizations on the MC convergence. We propose a numerical MLMC estimator, based on finite samples of numerical solutions with finite mean kinetic energy in a suitable function space and give sufficient conditions for mean-square convergence to a (generalized) moment of the statistical solution. We provide in particular error bounds for MLMC approximations of statistical solutions to the viscous Burgers equation in space dimension \(d=1\) and to the viscous, incompressible Navier-Stokes equations in space dimension \(d=2\) which are uniform with respect to the viscosity parameter. For a more detailed presentation and proofs we refer the reader to our work [Multilevel Monte Carlo approximations of statistical solutions of the Navier-Stokes equations. Research Rep. 2013-33, Seminar for Applied Mathematics, ETH Zürich (2013; doi:10.3929/ethz-a-010386317)].For the entire collection see [Zbl 1347.65003]. Cited in 2 Documents MSC: 76M35 Stochastic analysis applied to problems in fluid mechanics 65C05 Monte Carlo methods 35Q30 Navier-Stokes equations 65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs 76D06 Statistical solutions of Navier-Stokes and related equations 76M12 Finite volume methods applied to problems in fluid mechanics Keywords:multilevel Monte Carlo method; Navier-Stokes equations; statistical solutions; finite volume Software:ALSVID-UQ PDFBibTeX XMLCite \textit{A. Barth} et al., Springer Proc. Math. Stat. 163, 209--227 (2016; Zbl 1382.76209) Full Text: DOI References: [1] ALSVID-UQ. Version 3.0. http://www.sam.math.ethz.ch/alsvid-uq [2] Abdulle, A., Barth, A., Schwab, Ch.: Multilevel Monte Carlo methods for stochastic elliptic multiscale PDEs. 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