Semigroup Splitting And Cubature Approximations For The Stochastic Navier-Stokes Equations

TL;DR

This paper develops high-order numerical methods, including splitting schemes and cubature on Wiener space, for approximating the distribution of solutions to stochastic 2D Navier-Stokes equations, with convergence rates and numerical validation.

Contribution

It introduces and analyzes high-order splitting and cubature methods for stochastic Navier-Stokes equations, providing convergence rates and numerical evidence of effectiveness.

Findings

Convergence rates depend strongly on spectral discretization degree N.

Methods achieve convergence with appropriately chosen time steps.

Numerical simulations confirm practical applicability.

Abstract

Approximation of the marginal distribution of the solution of the stochastic Navier-Stokes equations on the two-dimensional torus by high order numerical methods is considered. The corresponding rates of convergence are obtained for a splitting scheme and the method of cubature on Wiener space applied to a spectral Galerkin discretisation of degree $N$. While the estimates exhibit a strong $N$ dependence, convergence is obtained for appropriately chosen time step sizes. Results of numerical simulations are provided, and confirm the applicability of the methods.