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

Numerical and convergence analysis of the stochastic Lagrangian averaged Navier-Stokes equations

The primary emphasis of this work is the development of a finite element based space-time discretization for solving the stochastic Lagrangian averaged Navier-Stokes (LANS-$α$) equations of incompressible fluid turbulence with multiplicative random forcing, under nonperiodic boundary conditions within a bounded polygonal (or polyhedral) domain of R^d , d $\in$ {2, 3}. The convergence analysis of a fully discretized numerical scheme is investigated and split into two cases according to the spacial scale $α$, namely we first assume $α$ to be controlled by the step size of the space discretizatio…

确定性与随机三维Navier--Stokes和Euler方程的非唯一遍历性

Non-unique ergodicity for deterministic and stochastic 3D Navier--Stokes and Euler equations

本文证明三维Navier--Stokes和Euler方程在确定性与随机加性噪声下存在无穷多个平稳及遍历平稳解，通过随机凸积分方法获得，并建立粘性消失极限关系。

An Implicit Euler Scheme with Non-uniform Time Discretization for Heat Equations with Multiplicative Noise

We present an algorithm for solving stochastic heat equations, whose key ingredient is a non-uniform time discretization of the driving Brownian motion $W$. For this algorithm we derive an error bound in terms of its number of evaluations of one-dimensional components of $W$. The rate of convergence depends on the spatial dimension of the heat equation and on the decay of the eigenfunctions of the covariance of $W$. According to known lower bounds, our algorithm is optimal, up to a constant, and this optimality cannot be achieved by uniform time discretizations.

随机超粘性Navier-Stokes方程的无粘极限与$\mathbb R^2$上Euler方程的不变测度

Inviscid Limit of the Stochastic Hyperviscous Navier-Stokes Equations and Invariant Measures for the Euler Equations in $\mathbb R^2$

通过构造带随机噪声的超粘性Navier-Stokes方程的马尔可夫平稳过程，证明其无粘极限收敛到Euler方程的路径平稳解，从而得到Euler方程的不变测度及其矩估计。

Euler time discretization of Backward Doubly SDEs and Application to Semilinear SPDEs

This paper investigates a numerical probabilistic method for the solution of some semilinear stochastic partial differential equations (SPDEs in short). The numerical scheme is based on discrete time approximation for solutions of systems of decoupled forward-backward doubly stochastic differential equations. Under standard assumptions on the parameters, the convergence and the rate of convergence of the numerical scheme is proven. The proof is based on a generalization of the result on the path regularity of the backward equation.

Stochastically Symplectic Maps and Their Applications to Navier-Stokes Equation

Poincare's invariance principle for Hamiltonian flows implies Kelvin's principle for solution to Incompressible Euler Equation. Iyer-Constantin Circulation Theorem offers a stochastic analog of Kelvin's principle for Navier-Stokes Equation. Weakly symplectic diffusions are defined to produce stochastically symplectic flows in a systematic way. With the aid of symplectic diffusions, we produce a family of martigales associated with solutions to Navier-Stokes Equation that in turn can be used to prove Iyer-Constantin Circulation Theorem. We also review some basic facts in symplectic and contact …

Inviscid limit for Stochastic Navier-Stokes Equations under general initial conditions

We consider in a smooth and bounded two dimensional domain the convergence in the $L^2$ norm, uniformly in time, of the solution of the stochastic Navier-Stokes equations with additive noise and no-slip boundary conditions to the solution of the corresponding Euler equations. We prove, under general regularity on the initial conditions of the Euler equations, that assuming the dissipation of the energy of the solution of the Navier-Stokes equations in a Kato type boundary layer, then the inviscid limit holds.

On strong $L^2$ convergence of time numerical schemes for the stochastic 2D Navier-Stokes equations

We prove that some discretization schemes for the 2D Navier-Stokes equations 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 NS 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…

Determining White Noise Forcing From Eulerian Observations in the Navier Stokes Equation

The Bayesian approach to inverse problems is of paramount importance in quantifying uncertainty about the input to and the state of a system of interest given noisy observations. Herein we consider the forward problem of the forced 2D Navier Stokes equation. The inverse problem is inference of the forcing, and possibly the initial condition, given noisy observations of the velocity field. We place a prior on the forcing which is in the form of a spatially correlated temporally white Gaussian process, and formulate the inverse problem for the posterior distribution. Given appropriate spatial re…