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 ⊂ ℝd, d = 1, 2 with no-slip or periodic boundary conditions on the boundary ∂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 Barth et al. (Multilevel Monte Carlo approximations of statistical solutions of the Navier-Stokes equations, 2013, [6]).
CITATION STYLE
Barth, A., Schwab, C., & Šukys, J. (2016). Multilevel monte carlo simulation of statistical solutions to the navier-stokes equations. In Springer Proceedings in Mathematics and Statistics (Vol. 163, pp. 209–227). Springer New York LLC. https://doi.org/10.1007/978-3-319-33507-0_8
Mendeley helps you to discover research relevant for your work.