Numerical analysis of 2D Navier--Stokes equations with additive stochastic forcing

TL;DR

This paper develops and analyzes a numerical discretization method for the 2D stochastic Navier--Stokes equations with additive noise, achieving strong convergence rates up to 1, extending previous results from linear SPDEs to nonlinear equations.

Contribution

It introduces a novel discretization approach for the nonlinear stochastic Navier--Stokes equations and proves strong convergence rates up to 1, a significant advancement over prior linear-only results.

Findings

Achieves strong convergence rate up to 1 in probability for discretized equations.

Extends convergence results from linear SPDEs to nonlinear stochastic Navier--Stokes equations.

Provides a framework for numerical analysis of 2D stochastic fluid dynamics equations.

Abstract

We propose and study a temporal, and spatio-temporal discretisation of the 2D stochastic Navier--Stokes equations in bounded domains supplemented with no-slip boundary conditions. Considering additive noise, we base its construction on the related nonlinear random PDE, which is solved by a transform of the solution of the stochastic Navier--Stokes equations. We show strong rate (up to) $1$ in probability for a corresponding discretisation in space and time (and space-time). Convergence of order (up to) 1 in time was previously only known for linear SPDEs.