Comparison Of Different Noise Forcings, Regularization Of Noise, And Optimal Control For The Stochastic Navier-Stokes Equation

摘要

Stochastic Navier-Stokes equations have been widely applied in various computational fluid dynamics (CFD) fields in recent years. It can be considered as another milestone in CFD. Our work focuses on exploring some theoretical and numerical properties of the stochastic Navier-Stokes equations and related optimal control problems. In particular, we consider: a numerical comparison of solutions of the stochastic Navier-Stokes equations perturbed by a large range of random noises in time and space; effective Martingale regularized methods for the stochastic Navier-Stokes equations with additive noises; and the stochastic Navier-Stokes equations constrained stochastic boundary optimal control problems.;We systemically provide numerical simulation methods for the stochastic Navier-Stokes equations with different types of noises. The noises are classified as colored or white based on their autocovariance functions. For each type of noise, we construct a representation and a simulation method. Numerical examples are provided to illustrate our schemes. Comparisons of the influence of different noises on the solution of the Navier-Stokes system are presented.;To improve the simulation accuracy, we impose a Martingale correction regularized method for the stochastic Navier-Stokes equations with additive noise. The original systems are split into two parts, a linear stochastic Stokes equations with Martingale solution and a stochastic modified Navier-Stokes equations with smoother noise. In addition, a negative fractional Laplace operator is introduced to regularize the noise term. Stability and convergence of the path-wise modified Navier-Stokes equations are proved. Numerical simulations are provided to illustrate our scheme. Comparisons of non-regularized and regularized noises for the Navier-Stokes system are presented to further demonstrate the efficiency of our numerical scheme.;As a consequence of the above work, we consider a stochastic optimal control problem constrained by the Navier-Stokes equations with stochastic Dirichlet boundary conditions. Control is applied only on the boundary and is associated with reduced regularity, compared to interior controls. To ensure the existence of a solution and the efficiency of numerical simulations, the stochastic boundary conditions are required to belong almost surely to H1. To simulate the system, state solutions are approximated using the stochastic collocation finite element approach, and sparse grid techniques are applied to the boundary random field. One-shot optimality systems are derived from Lagrangian functionals. Numerical simulations are then made, using a combination of Monte Carlo methods and sparse grid methods, which demonstrate the efficiency of the algorithm.