Mean square temporal error estimates for the two-dimensional stochastic Navier–Stokes equations with transport noise

We study the two-dimensional Navier–Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time discretization 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 Itô-type noise.