Numerical analysis of the stochastic Navier-Stokes equations

The developments over the last five decades concerning numerical discretisations of the incompressible Navier--Stokes equations have lead to reliable tools for their approximation: those include stable methods to properly address the incompressibility constraint, stable discretisations to account for convection dominated problems, efficient time (splitting) methods, and methods to tackle their nonlinear character. While these tools may successfully be applied to reliably simulate even more complex fluid flow PDE models, their understanding requires a fundamental revision in the case of stochastic fluid models, which are gaining increased importance nowadays. This work motivates and surveys optimally convergent numerical methods for the stochastic Stokes and Navier--Stokes equations that were obtained in the last decades. Furtheremore, we computationally illustrate the failure of some of those methods from the deterministic setting, if they are straight-forwardly applied to the stochastic case. In fact, we explain why some of these deterministic methods perform sub-optimally by highlighting crucial analytical differences between the deterministic and stochastic equations -- and how modifications of the deterministic methods restore their optimal performance if they properly address the probabilistic nature of the stochastic problem. Next to the numerical analysis of schemes, we propose a general benchmark of prototypic fluid flow problems driven by different types of noise to also compare new algorithms by simulations in terms of complexities, efficiencies, and possible limitations. The driving motivation is to reach a better comparison of simulations for new schemes in terms of accuracy and complexities, and to also complement theoretical performance studies for restricted settings of data by more realistic ones.