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 $L^2(\Omega, \mathbb{P})$ 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.
| Original language | English |
|---|---|
| Pages (from-to) | 433–470 |
| Number of pages | 38 |
| Journal | Stochastic Partial Differential Equations: Analysis and Computations |
| Volume | 2 |
| Issue number | 4 |
| Early online date | 30 Oct 2014 |
| DOIs | |
| Publication status | Published - Dec 2014 |
Keywords
- Splitting up methods
- Navier–Stokes equations
- Hydrodynamical models
- Stochastic PDEs
- Strong convergence
- Speed of convergence in probability