Numerical investigation of the parabolic mixed-derivative diffusion equation via alternating direction implicit methods
Date
2013-08-07
Authors
Sathinarain, Melisha
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Abstract
In this dissertation, we investigate the parabolic mixed derivative diffusion equation modeling
the viscous and viscoelastic effects in a non-Newtonian viscoelastic fluid. The model is
analytically considered using Fourier and Laplace transformations. The main focus of the
dissertation, however, is the implementation of the Peaceman-Rachford Alternating Direction
Implicit method. The one-dimensional parabolic mixed derivative diffusion equation
is extended to a two-dimensional analog. In order to do this, the two-dimensional analog
is solved using a Crank-Nicholson method and implemented according to the Peaceman-
Rachford ADI method. The behaviour of the solution of the viscoelastic fluid model is
analysed by investigating the effects of inertia and diffusion as well as the viscous behaviour,
subject to the viscosity and viscoelasticity parameters. The two-dimensional parabolic diffusion
equation is then implemented with a high-order method to unveil more accurate
solutions. An error analysis is executed to show the accuracy differences between the numerical
solutions of the general ADI and high-order compact methods. Each of the methods
implemented in this dissertation are investigated via the von-Neumann stability analysis to
prove stability under certain conditions.
Description
A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment of the requirements for the degree of Master of Science, May 14, 2013.