Splitting up method for the 2D stochastic Navier–Stokes equations

Abstract

In this paper, we deal with the convergence of an iterative scheme for the 2-D stochastic Navier-Stokes Equations on the torus suggested by the Lie-Trotter product formulas for stochastic differential equations of parabolic type. The stochastic system is split into two problems which are simpler for numerical computations. An estimate of the approximation error is given for periodic boundary conditions. In particular, we prove that the strong speed of the convergence in probability is almost 1/2. This is shown by means of an L 2 (Ω, P) convergence localized on a set of arbitrary large probability. The assumptions on the diffusion coefficient depend on the fact that some multiple of the Laplace operator is present or not with the multiplicative stochastic term. Note that if one of the splitting steps only contains the stochastic integral, then the diffusion coefficient may not contain any gradient of the solution.

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

“…Note that for large ν, γ approaches 1 2 . For the splitting scheme, the strong speed of convergence we obtain in this paper is better than that of the convergence in probability proven in [3], although not polynomial; it is of the form c exp(−C √ N ). In this paper, we only deal with time discretization, unlike in [9] where a space-time discretization is studied.…”

“…To explain the method, the paper will deal with two different algorithms: the splitting scheme used in [3] and the implicit Euler schemes used in [9]. In the case of a diffusion coefficient G with linear growth conditions, which may depend on the solution and its gradient for the Euler schemes, we prove that the speed of convergence of both schemes is any negative power of the logarithm of the time mesh T N when the initial condition belongs to W 1,2 and is divergence free.…”

“…We formulate the assumptions on the noise. The splitting scheme is described in Section 3 and various moments estimates previously used in [3] are recalled. The strategy for proving the strong speed of convergence is described and explained in details.…”

A note on the stochastic Ericksen-Leslie equations for nematic liquid crystals

“…Throughout this paper we denote by L p pOq and W m,p pOq, p P r1, 8s, m P N, the Lebesgue and Sobolev spaces of real valued functions defined on O, see e.g. the monograph [48] by Temam (compare [3]). The corresponding spaces of R d (or some cases R 3 )-valued functions, will be denoted by the black-board fonts, e.g.…”

Layer methods for stochastic Navier–Stokes equations using simplest characteristics

“…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. They used splitting ideas similar to the ones considered in [17,18] for linear parabolic stochastic partial di¤erential equations (SPDE).…”