Approximation of deterministic and stochastic Navier-Stokes equations in vorticity-velocity formulation

TL;DR

This paper introduces a new time discretization method for the Navier-Stokes equations in vorticity-velocity form, providing convergence results for both deterministic and stochastic cases, including the first mean-square convergence order for stochastic equations.

Contribution

It proposes a novel linear parabolic approximation scheme for Navier-Stokes equations in vorticity-velocity form with proven divergence-free property and convergence, extending to stochastic equations with additive noise.

Findings

First-order divergence-free approximation scheme.

Proven mean-square convergence order for stochastic Navier-Stokes.

Extension of results to stochastic equations with additive noise.

Abstract

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in linear parabolic equations for vorticity. Probabilistic representations for solutions of these linear equations are given. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot--Savart-type law. We show that the approximation is divergent free and of first order. The results are extended to two-dimensional stochastic Navier-Stokes equations with additive noise, where, in particular, we prove the first mean-square convergence order of the vorticity approximation.