Overlapping grid spectral collocation methods for nonlinear differential equations modelling fluid flow problems.
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Date
2020
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Abstract
The focus of this thesis is on computational grid-manipulation to enhance the accuracy, convergence
and computational efficiency of spectral collocation methods for the solution of differential
equations in fluid mechanics. The need to develop highly accurate, convergent and computationally
efficient numerical techniques for solving nonlinear problems is an ever-recurring theme in
numerical mathematics. Spectral methods have been shown in the literature to be more accurate
and efficient than some common numerical methods, such as finite difference methods. However,
their accuracy deteriorates as the computational domain increases and when the number of grid
points reaches a certain critical value. The spectral collocation algorithm produces dense matrix
equations, for which there is no known efficient solution method. These deficiencies necessitate
the development of spectral techniques that produce less dense matrix equations using fewer grid
points. This thesis presents a new improvement in spectral collocation methods with particular
application to nonlinear differential equations that model problems arising in fluid mechanics. The
improvement described in this thesis requires the use of overlapping grids when descritizing the
solution domain for Chebyshev spectral collocation method. The thesis is presented in two related
subdivisions. In Part A, the overlapping grid approach is used only in space variable when solving
nonlinear ordinary and partial differential equations. Subsequently, the overlapping grid approach
is used in both the space and time variables in the solution of partial differential equations.
This thesis is also devoted to analysing solutions of fluid flow models through various practical
geometries with particular interest in non-Newtonian fluid flows. The physics of these fluid flows
is studied through parametric studies on the effects of diverse thermophysical parameters on the
fluid properties, changes in shear stresses, and heat and mass transport. Key findings, are inter
alia, that the overlapping multi-domain spectral techniques are computationally efficient, produce
stable and accurate results using a small number of grid points in each subinterval and in the
entire computational domain. Using the overlapping grids yields less dense coefficient matrices
that invert easily. Changes in thermophysical parameters has significant consequences for the fluid
properties, and heat and mass transfer processes.
Description
Doctoral Degree. University of KwaZulu-Natal, Durban.