Splitting up method for the 2D stochastic Navier-Stokes equations

TL;DR

This paper analyzes the convergence of a splitting scheme for the 2D stochastic Navier-Stokes equations, demonstrating a strong convergence rate of nearly 1/2 under certain conditions, facilitating numerical solutions.

Contribution

It introduces a Lie-Trotter based splitting method for the stochastic Navier-Stokes equations and provides convergence estimates including error bounds and rates.

Findings

Strong convergence rate is almost 1/2.

Error estimates are provided for the splitting scheme.

Convergence is localized in probability on large sets.

Abstract

In this paper, we deal with the convergence of an iterative scheme for the 2-D stochastic Navier-Stokes Equations on the torus suggested by the Lie-Trotter product formulas for stochastic differential equations of parabolic type. The stochastic system is split into two problems which are simpler for numerical computations. An estimate of the approximation error is given either with periodic boundary conditions. In particular, we prove that the strong speed of the convergence in probability is almost $1/2$. This is shown by means of an $L^2(\Omega,P)$ convergence localized on a set of arbitrary large probability. The assumptions on the diffusion coefficient depend on the fact that some multiple of the Laplace operator is present or not with the multiplicative stochastic term. Note that if one of the splitting steps only contains the stochastic integral, then the diffusion coefficient may not contain any gradient of the solution.