Semigroup Splitting and Cubature Approximations for the Stochastic Navier–Stokes Equations

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.

Strong $L^2$ convergence of time numerical schemes for the stochastic two-dimensional Navier–Stokes equations

“…In [3], the authors used a splitting method, based on the Lie-Trotter formula, proving again some rate of convergence in probability of the numerical scheme. In [12], P.…”

Cubature methods for stochastic (partial) differential equations in weighted spaces

“…We extended the functional analytic framework of Röck-ner and Sobol [29], used for the numerical analysis of stochastic evolution equations in Dörsek [10] and Dörsek and Teichmann [11,12], to more general characteristics through a flexible formulation of directional derivatives in weighted spaces. This setting was then used to prove optimal rates of convergence of cubature schemes for more general equations.…”

“…We recall the following definition of spaces of functions with controlled growth, see also Röckner and Sobol [29] for their use in the construction of the solution of martingale problems in infinite dimension, and Dörsek [9,10] and Dörsek and Teichmann [11,12] for their application to the analysis of splitting schemes for stochastic partial differential equations. …”

“…Instead, we follow a similar idea as was applied to the stochastic Navier-Stokes equations in Dörsek [10]. We extend the results of Bayer and Teichmann [2], where strong conditions are imposed on the vector fields, to more general coefficients and test functions.…”

Layer methods for stochastic Navier–Stokes equations using simplest characteristics

“…In [6,4] implicit and semi-discrete Euler time and …nite element based space-time discretizations are studied, convergence is proved in the mean-square (strong) sense. The work [14] deals with a time-splitting scheme combined with a Galerkin approximation in the space variable for SNSE exploiting the semi-group and cubature techniques, a weak convergence is proved for the proposed method. In [3] the authors consider a method based on splitting SNSE in a deterministic NSE and stochastic Stokes equation, they prove convergence in the mean-square sense and in probability of the method.…”