Mean Square Temporal error estimates for the 2D stochastic Navier-Stokes equations with transport noise

TL;DR

This paper establishes a mean square convergence rate of order 1/2 for the time discretization of the 2D stochastic Navier-Stokes equations with transport noise, supported by theoretical estimates and numerical simulations.

Contribution

It provides the first mean square error convergence rate for the stochastic Navier-Stokes equations with transport noise, using uniform-in-probability estimates and structural noise analysis.

Findings

Convergence rate of order 1/2 in mean square error for time discretization.

Numerical simulations confirm the theoretical convergence rate.

Comparison of energy profiles with additive and multiplicative noise cases.

Abstract

We study the 2D Navier-Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time-discretisation showing a convergence rate of order (up to) 1/2. It holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier-Stokes equations was only known with respect to convergence in probability. Our result is based on uniform-in-probability estimates for the continuous as well as the time-discrete solution exploiting the particular structure of the noise. Eventually, we perform numerical simulations for the corresponding problem on bounded domains with no-slip boundary conditions. They suggest the same convergence rate as proved for the periodic problem hinging sensitively on the compatibility of the data. We also compare the energy profiles with those for corresponding problems with additive or multiplicative It\^o-type noise.