Defense titled "Fast and Accurate Methods for Predicting Fluid Flows". Fluids in motion exhibit complex behaviors that are difficult to predict yet critical to applications such as renewable energy and weather forecasting. Since fluid flows share common structures, such as boundary layers and vortices, numerical simulations can accurately predict behavior across a wide range of applications. The Navier–Stokes equations form the foundational mathematical model for nearly all fluid flows. For incompressible flows, pressure acts as a Lagrange multiplier enforcing the divergence-free constraint on velocity, which results in a coupled and computationally challenging system
Due to the chaotic nature of fluid dynamics, tiny errors in initial conditions can amplify rapidly. It leads to finite predictability horizons. To address uncertainties in problem data, ensemble methods run simulations over a range of initial and boundary conditions. While these methods improve statistical prediction, they are computationally expensive, requiring the solution of J separate systems at each time step for an ensemble of size J.
We first developed a penalty-based ensemble method that reduces computational cost by reusing a shared coefficient matrix across ensemble members and relaxing the incompressibility condition. This approach decouples pressure from velocity, eliminates J pressure variables, and lowers memory and operation costs while maintaining accuracy.
However, the performance of penalty methods is sensitive to the choice of the penalty parameter ϵ, and no effective á priori formula exists. The second project extends Xie's adaptive penalty method for the Stokes problem to the nonlinear, time-dependent Navier–Stokes equations. The method chooses ϵ elementwise, adapting to the local divergence error ∇⋅u^h. We provide a comprehensive theoretical analysis to validate it. 
Direct Numerical Simulation (DNS) is often impractical due to its demand for fine-grained spatial and temporal resolution. Unsteady Reynolds-averaged Navier Stokes (URANS) models offer a practical alternative, available in 0-equation, 1-equation, and 2-equation forms, with increasing complexity and predictive power. The third project studies a 1/2-equation model that simplifies the standard 1-equation turbulence model. We replace the partial differential equation for turbulent kinetic energy k(x,t) with its spatial average k(t). This approach reduces complexity to that of a 0-equation model while retaining the accuracy of the 1-equation formulation. The last project presents the numerical analysis of this 1/2-equation model, including proofs of uniqueness for strong solutions, as well as stability, convergence, and error estimates.
Committee Chair and Advisor: Dr. William Layton