Computational and numerical analysis of differential equations using spectral based collocation method.
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Date
2019
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Abstract
In this thesis, we develop accurate and computationally efficient spectral collocation-based methods,
both modified and new, and apply them to solve differential equations. Spectral collocation-based
methods are the most commonly used methods for approximating smooth solutions of differential
equations defined over simple geometries. Procedurally, these methods entail transforming the gov
erning differential equation(s) into a system of linear algebraic equations that can be solved directly.
Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported
in the literature, researchers often transform their models to reduce the number of variables or
narrow them down to problems with fewer dimensions. Such a process is accomplished by making
a series of assumptions that limit the scope of the study. To address this deficiency, the present
study explores the development of numerical algorithms for solving ordinary and partial differential
equations defined over simple geometries. The solutions of the differential equations considered are
approximated using interpolating polynomials that satisfy the given differential equation at se
lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the
computational domain is particularly emphasized as it plays a key role in determining the number
of grid points that are used; a feature that dictates the accuracy and the computational expense of
the spectral method. To solve differential equations defined on large computational domains much
effort is devoted to the development and application of new multidomain approaches, based on
decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time
interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con
firms the superiority of these multiple domain techniques in terms of accuracy and computational
efficiency over the single domain approach when applied to problems defined over large domains.
The structure of the thesis indicates a smooth sequence of constructing spectral collocation method
algorithms for problems across different dimensions. The process of switching between dimensions
is explained by presenting the work in chronological order from a simple one-dimensional problem
to more complex higher-dimensional problems. The preliminary chapter explores solutions of or
dinary differential equations. Subsequent chapters then build on solutions to partial differential
i
equations in order of increasing computational complexity. The transition between intermediate
dimensions is demonstrated and reinforced while highlighting the computational complexities in
volved. Discussions of the numerical methods terminate with development and application of a
new method namely; the trivariate spectral collocation method for solving two-dimensional initial
boundary value problems. Finally, the new error bound theorems on polynomial interpolation are
presented with rigorous proofs in each chapter to benchmark the adoption of the different numerical
algorithms. The numerical results of the study confirm that incorporating domain decomposition
techniques in spectral collocation methods work effectively for all dimensions, as we report highly
accurate results obtained in a computationally efficient manner for problems defined on large do
mains. The findings of this study thus lay a solid foundation to overcome major challenges that
numerical analysts might encounter.
Description
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.