Date of Award

2021

Document Type

Open Access Master's Thesis

Degree Name

Master of Science in Mathematical Sciences (MS)

Administrative Home Department

Department of Mathematical Sciences

Advisor 1

Alexander Labovsky

Committee Member 1

Jiguang Sun

Committee Member 2

Zhengfu Xu

Abstract

A sizeable proportion of the work in this thesis focuses on a new turbulence model, dubbed ADC (the approximate deconvolution model with defect correction). The ADC is improved upon using spectral deferred correction, a means of constructing a higher order ODE solver. Since both the ADC and SDC are based on a predictor-corrector approach, SDC is incorporated with essentially no additional computational cost. We will show theoretically and using numerical tests that the new scheme is indeed higher order in time than the original, and that the benefits of defect correction, on which the ADC is based, are preserved.

The final two chapters in this thesis focus on a two important numerical difficulties arising in fluid flow modeling: poor mass-conservation and possible non-physical oscillations. We show that grad-div stabilization, previously assumed to have no effect on the target quantities of the test problem used, can significantly alter the results even on standard benchmark problems. We also propose a work-around and verify numerically that it has promise. Then we investigate two different formulations of Crank-Nicolson for the Navier-Stokes equations. The most attractive implementation, second order accurate for both velocity and pressure, is shown to introduce non-physical oscillations. We then propose two options which are shown to avoid the poor behavior

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