Approximation of deterministic and stochastic Navier-Stokes equations in vorticity-velocity formulation
This provides a new numerical method for solving Navier-Stokes equations, particularly useful for stochastic formulations, but the improvement is incremental over existing splitting methods.
The authors propose a first-order time discretization for the Navier-Stokes equations in vorticity-velocity form, freezing velocity on subintervals to obtain linear parabolic equations for vorticity. They prove first-order convergence for the deterministic case and first mean-square convergence order for the stochastic case with additive noise.
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.