On strong $L^2$ convergence of time numerical schemes for the stochastic 2D Navier-Stokes equations

TL;DR

This paper demonstrates strong $L^2$ convergence of various time discretization schemes for the stochastic 2D Navier-Stokes equations, refining previous probabilistic convergence results by establishing almost sure convergence with explicit rates.

Contribution

It proves strong $L^2$ convergence for implicit, semi-implicit, and splitting schemes for stochastic 2D Navier-Stokes equations, using exponential moment estimates and localized scheme analysis.

Findings

Convergence speed depends on diffusion coefficient and viscosity.

Fully implicit, semi-implicit, and splitting schemes all converge strongly.

Refines previous probabilistic convergence results to strong $L^2$ convergence.

Abstract

We prove that some discretization schemes for the 2D Navier-Stokes equations subject to a random perturbation converge in $L^2(\Omega)$. This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic NS equations and convergence of a localized scheme, we can prove strong convergence of fully implicit and semi-implicit time Euler discretizations, and of a splitting scheme. The speed of the $L^2(\Omega)$-convergence depends on the diffusion coefficient and on the viscosity parameter.