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

@article{Bessaih2018StrongC, title={Strong \$L^2\$ convergence of time numerical schemes for the stochastic two-dimensional Navier–Stokes equations}, author={Hakima Bessaih and Annie Millet}, journal={IMA Journal of Numerical Analysis}, year={2018} }

We prove that some time discretization schemes for the two-dimensional Navier–Stokes equations on the torus subject to a random perturbation converge in $L^2(\varOmega )$. This refines previous results that established the convergence only 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 semiimplicit temporal…

## 11 Citations

Convergence rates for the numerical approximation of the 2D stochastic Navier-Stokes equations

- Mathematics, Computer ScienceNumerische Mathematik
- 2021

This work establishes convergence rates for a finite-element based space-time approximation with respect to convergence in probability and provides linear convergence in space and convergence of order (almost) 1/2 in time.

Numerical Analysis of Fully discrete Finite element Methods for the stochastic Navier-Stokes Equations with Multiplicative Noise

- Mathematics
- 2021

Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations

- MathematicsArXiv
- 2019

The main result of this article establishes strong convergence rates on the whole probability space for explicit space-time discrete numerical approximations for a class of stochastic evolution…

Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities

- Mathematics, Computer Science
- 2019

This work proves that full- Discrete exponential Euler and full-discrete linear-implicit Euler approximations diverge strongly and numerically weakly in the case of stochastic Allen-Cahn equations.

Strong convergence rate of a full discretization for stochastic Cahn--Hilliard equation driven by space-time white noise

- Mathematics
- 2018

In this article, we consider the stochastic Cahn--Hilliard equation driven by space-time white noise. We discretize this equation by using a spatial spectral Galerkin method and a temporal…

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

- Mathematics, Computer ScienceDiscrete & Continuous Dynamical Systems - B
- 2019

This work introduces and analyzes an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise, and proves boundedness of moments of the numerical solution.

Analysis of Chorin-Type Projection Methods for the Stochastic Stokes Equations with General Multiplicative Noises

- Computer ScienceArXiv
- 2020

It is proved that all spatial error constants contain a growth factor, which explains the deteriorating performance of the standard Chorin scheme when $k\to 0$ and the space mesh size is fixed as observed earlier in the numerical tests of [9].

Point vortex approximation for 2D Navier–Stokes equations driven by space-time white noise

- Mathematics, Physics
- 2019

Strong $$L^2$$ convergence of time Euler schemes for stochastic 3D Brinkman–Forchheimer–Navier–Stokes equations

- MathematicsStochastics and Partial Differential Equations: Analysis and Computations
- 2022

Space-time Euler discretization schemes for the stochastic 2D Navier-Stokes equations

- MathematicsStochastics and Partial Differential Equations: Analysis and Computations
- 2021

We prove that the implicit time Euler scheme coupled with finite elements space discretization for the 2D Navier-Stokes equations on the torus subject to a random perturbation converges in…

## References

SHOWING 1-10 OF 31 REFERENCES

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

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2012

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 and convergence is obtained for appropriately chosen time step sizes.

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

- Mathematics
- 2013

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…

Rates of Convergence for Discretizations of the Stochastic Incompressible Navier-Stokes Equations

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2012

It turns out that it is the interaction of Lagrange multipliers with the stochastic forcing in the scheme which limits the accuracy of general discretely LBB-stable space discretizations, and strategies to overcome this problem are proposed.

Galerkin Approximations for the Stochastic Burgers Equation

- MathematicsSIAM J. Numer. Anal.
- 2013

The main novelty in this article is the estimation of the difference of the finite-dimensional Galerkin approximations and of the solution of the infinite-dimensional SPDE uniformly in space, instead of the usual Hilbert space estimates in the $L^2$-topology, that were shown before.

Numerical Approximations of Stochastic Differential Equations With Non-globally Lipschitz Continuous Coefficients

- Mathematics
- 2012

Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, momentsof the computationally…

Large Deviations and the Zero Viscosity Limit for 2D Stochastic Navier-Stokes Equations with Free Boundary

- MathematicsSIAM J. Math. Anal.
- 2012

Using a weak convergence approach, we prove a LPD for the solution of 2D stochastic Navier Stokes equations when the viscosity converges to 0 and the noise intensity is multiplied by the square root…

On the discretization in time of parabolic stochastic partial differential equations

- Mathematics, Computer ScienceMonte Carlo Methods Appl.
- 2001

Although the author is not able in this case to compute a pathwise order of the approximation, the weaker notion of order in probability is introduced and the results of the globally Lipschitz case are generalized.

Inviscid Large deviation principle and the 2D Navier Stokes equations with a free boundary condition

- Mathematics
- 2010

Using a weak convergence approach, we prove a LPD for the solution of 2D stochastic Navier Stokes equations when the viscosity converges to 0 and the noise intensity is multiplied by the square root…

Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities

- MathematicsIMA Journal of Numerical Analysis
- 2019

In this article we propose a new, explicit and easily implementable numerical method for approximating a class of semilinear stochastic evolution equations with non-globally Lipschitz continuous…

On the discretization in time of parabolic stochastic partial differential equations

- Mathematics, Computer Science
- 2001

In an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation are generalized.