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

TL;DR

This paper introduces a higher-order time discretization scheme for stochastic Navier--Stokes equations with additive noise, achieving strong convergence rate 1.5, supported by theoretical analysis and numerical simulations.

Contribution

It presents a novel higher-order discretization method based on reformulating the equations as a random PDE, improving convergence rates for stochastic Navier--Stokes equations.

Findings

Achieves strong convergence rate of 1.5 in probability.

Uses reformulation as a random PDE for improved discretization.

Numerical simulations confirm theoretical results.

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.