Doctoral Dissertations

Date of Award

12-1997

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Engineering Science

Major Professor

A. J. Baker

Committee Members

Jay Frankel, Joe Iannelli, Terry Miller

Abstract

Taylor weak statement (TWS) or Taylor-Galerkin finite element method approxi- mations were formulated for the advection-diffusion equation, the Burgers equation, and the incompressible Navier-Stokes equations. A new TWS formulation was developed for the shallow-water equations. A Fourier analysis was performed to compare phase ac- curacy and stability for linear basis finite element methods; a new Fourier analysis was also performed for quadratic and cubic basis TWS one-dimensional finite element meth- ods. A multiple-step linear basis TWS finite element method was derived and optimized for phase accuracy. A new modified equation analysis was developed in a general linear form, and the analysis was used to optimize two-dimensional TWS methods for phase ac- curacy and for stability. Phase-accurate TWS finite element methods were found effective in maintaining time accuracy in scalar field equations, but of limited value for the transient Navier-Stokes equation systems tested.

Stability-enhanced artificial diffusion TWS methods for the incompressible Navier- Stokes equations were systematically compared for the first time to a wide variety of modified finite element methods, including SUPG, exponential Petrov-Galerkin, SGM, hierarchical basis, least-squares, and monotone methods. The artificial diffusion meth- ods were tested on various two-dimensional advection-diffusion and incompressible laminar Navier-Stokes benchmark problems. In contrast to the phase-accurate methods, the stability-enhanced methods were found to effectively decrease grid requirements for high Reynolds number laminar flow benchmark problems, including the backward-facing step and cylinder-in-crossflow. The best performing methods were the exponential Petrov- Galerkin and streamline-diffusion (TWS or SUPG) methods. As expected, the artificial diffusion methods were found ineffective on a high Rayleigh number natural convection benchmark problems characterized by small magnitude of velocity.

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