Date of Award


Document Type


Degree Name

Master of Science


Department of Mathematics and Statistics

First Advisor

Benjamin F. Akers, PhD


This research introduces a fixed-point numerical approach for solving the steady-state Navier-Stokes (NS) equations on a finite two-dimensional (2D) domain. The steady-state interaction between a high energy laser beam and its surrounding fluid medium is important to researchers in the field of high energy laser beam propagation. The solutions to the steady-state Navier-Stokes equations provide a model for uncovering the steady-state behavior of the fluid medium, which is useful for the modeling of thermal blooming in laser beam propagation. Numerical solutions remain the only tenable option for solving the NS equations, wherein numerical speed and fidelity beget the utility of any such algorithm. The timing and accuracy results from the novel fixed-point algorithm are compared to a standard Newton solver, where the fixed-point algorithm implements a series of discrete Poisson solvers through successive fixed-point iterations of fluid velocity (u,v), pressure (p), and temperature (T) in a Boussinesq fluid model. The fixed-point scheme consistently proves superior in computational cost by converging after O(N2 log N2 ) flops compared to the O(N6) flops in the Newton Solver for a discrete N x N grid. We provide a proof for the convergence of small amplitude solutions, and discuss the relationship between fluid parameters (Re, Ri, Pe) and the existence of solutions as a function of laser intensity in a bifurcation analysis.

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