Multilevel Monte Carlo Simulation of Statistical Solutions to the Navier–Stokes Equations

Abstract

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 Barth et al. (Multilevel Monte Carlo approximations of statistical solutions of the Navier–Stokes equations, 2013, [6]).