A study on the random vortex methods and their related equations

Thesis

This thesis is mainly divided into three parts. In Chapter 3, we propose a simple yet powerful random vortex method to approximate the fluid flow dynamics. The idea is that we sample the initial vortices of the fluid flow over some Brownian fluid particles and then track their vortex dynamics. The weak convergence of the approximation from our scheme is shown. We also demonstrate its effectiveness through several numerical experiments.

Chapter 4 establishes a closed stochastic system which is equivalent to the Navier-Stokes equations for incompressible flows. This system consists of a forward stochastic differential equation and a ordinary functional differential equation, which reveals the vortex dynamics for a viscous fluid flow. We investigate two tools to derive the system: the duality of pinned diffusion measure and a forward type Feynman-Kac formula for nonlinear parabolic equations. We also conduct numerical simulations based on the system.

In Chapter 5, the random vortex system proposed in Chapter 4 is generalized to a class of stochastic differential equations of the random vortex type (RVSDEs). We investigate various problems on RVSDEs, such as existence and uniqueness theories, large deviation principles, and numerical methods, under certain nice assumptions.

Authors: Zhang, Y

• DOI: 10.5287/ora-zbwvd6xkk
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: stochastic differential equations; stochastic analysis; random vortex dynamics; Feynman–Kac formula; Navier–Stokes equations
• Subjects: Fluid dynamics; Navier-Stokes equations--numerical solutions; Stochastic analysis