Abstract
In this dissertation, we present sixth-order compact finite difference methodology for
solving linear and nonlinear second- and third-order ordinary differential equations
subject to multi-point and Robin type boundary conditions. Compact finite difference
methods are known for their high accuracy with low computational cost and
implementation complexity. A review of existing literature on compact finite difference
methods shows that, second- and third-order ordinary differential equations subject to
multi-point and Robin type boundary conditions respectively, have not been solved using
compact finite difference methods.
The sixth-order accurate compact finite difference schemes are derived for the interior
and boundary points. Therefore, the methodology used maintains the accuracy of the
schemes in the interior and at the boundaries of a given domain. Previous studies typically
use forward and backward difference approximation to approximate first order derivative
at the boundaries for boundary value problems with Neumann and Robin type boundary
conditions. This approach compromises the accuracy of the entire scheme. Hence, our
proposed approach is superior compared to the classical approach.
For nonlinear ordinary differential equations, a popular linearization technique
referred to as the quasi-linearization technique is used to reduce nonlinear differential
equations into a sequence of linear differential equations prior to applying the proposed
higher-order compact finite difference schemes.
Through solving multiple boundary value problems found in the literature, numerical
experiments demonstrate the high accuracy of the derived schemes in comparison with
exact solutions and established methods.
An introduction to boundary value problems and various types of commonly seen
boundary conditions is provided. Some of the characteristics of commonly used
deterministic numerical methods for solving boundary value problems are reviewed.
An introduction to classical and higher-order compact finite difference methods is also
provided. Recent developments in numerical discretization techniques are also shared as
part of the literature review. Lastly, considerations for future work using higher-order
compact finite difference schemes are provided as part of the conclusion.