A Higher Order Discretization for the Stochastic Navier--Stokes equations with additive Noise

Abstract

We propose a new higher-order time discretization scheme for the stochastic Navier--Stokes equations with additive noise, where its velocity and pressure approximates converge at strong rate $1.5$ in probability. The construction rests on its reformulation as a random PDE for the transform $y = u- \Phi W$, and different higher order numerical quadrature rules for the diffusion and the drift part. The theoretical findings are supported by numerical simulations.