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

Abstract

We prove that some time discretization schemes for the 2D Navier-Stokes equations on the torus subject to a random perturbation converge in L 2 (Ω). This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier-Stokes equations and convergence of a localized scheme, we can prove strong convergence of fully implicit and semi-implicit time Euler discretizations, and of a splitting scheme. The speed of the L 2 (Ω)-convergence depends on the diffusion coefficient and on the viscosity parameter.2000 Mathematics Subject Classification. Primary 60H15, 60H35; Secondary 76D06, 76M35.

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

“…In this work, we introduce new schemes, based on a splitting (also referred to as splitting-up) strategy, see for instance [5,6,24,25] in the SPDE case, and [43,51] for the deterministic Allen-Cahn equation. Indeed, the solution of the ordinary differential equationż = z − z 3 has a known explicit expression.…”

“…The schemes X exact and X expo correspond to versions of the exponential Euler scheme, applied to the auxiliary equation (6). The scheme X imp is the standard linear implicit Euler scheme, applied to the auxiliary equation (6).…”

Mean-Square Approximation of Navier-Stokes Equations with Additive Noise in Vorticity-Velocity Formulation

“…[13]). In majority of papers on numerical approximation of SNSE [2][3][4][5][6]9] the case of multiplicative noise is considered. The NSE with additive noise deserves a special attention due to its interesting properties [13,16,18].…”

“…Since we work in the vorticity-velocity formulation and aimed at reaching a higher order of meansquare convergence for the method considered, we require higher spatial smoothness of the velocity in our proofs than e.g. in [3] where mean-square convergence of fully and semi-implicit Euler schemes from [6] for SNSE with multiplicative and additive noise in the velocity formulation was considered. In the case of an additive noise, the authors of [3] proved mean-square convergence with a polynomial rate (up to 1 4 ) in the time mesh.…”

A Fully Discrete Mixed Finite Element Method for the Stochastic Cahn–Hilliard Equation with Gradient-Type Multiplicative Noise

“…Other recent works for stochastic Navier-Stokes equations include an iterative splitting scheme which was proposed in [6], a strong convergence in probability was established in the 2-D case for the velocity approximation. In a very recent paper [7], the authors proposed another time-splitting scheme and proved its strong L 2 convergence for the velocity approximation. In [31] a posterior error estimates were studied for a fully discrete divergence-free finite element method for the 2-D stochastic Navier-Stokes equations, both upper and lower a posterior error bounds were established for the velocity approximation in the paper.…”