Abstract
In this thesis, we introduced numerical methods for solving the advection diffusion reaction
equations which are paramount in modeling complex problems in diverse scientific fields, such
as physics, engineering and applied mathematics. The modelled problem in general is governed
by partial differential equations. It is difficult, in some cases impossible to get analytical solution
to most of the partial differential equations that arise in mathematical models of physical
phenomena. The numerical methods were developed to intervene in solving this nature of problems
and they were proven to perform exceedingly well in solving difficult partial differential
equations. The study aim to introduce the Enhanced Unconditionally Positive Finite Difference
(EUPFD), develop the Higher order Unconditionally Positive Finite Difference (HUPFD), and
to introduce the Enhanced Higher order Unconditionally Positive Finite Difference (EHUPFD)
Methods to solve the linear, non-linear, and system of advection diffusion reaction equations.
The proper orthogonal decomposition technique is used on the UPFD and HUPFD to reduce the
degree of freedom that arise in the advection diffusion reaction equations. This study investigated
the performance of the proposed methods by checking the stability and consistency, which
are necessary for confirming a well developed numerical methods. The absolute error, computational
time and convergence rate for the developed methods were calculated and compared
against other results obtained by the Crank-Nicolson, and non-standard finite difference methods.
A comparison of solutions obtained by the exact and proposed methods is also performed
for validation purposes, and is considered to mean that the proposed methods are effective in
solving the advection diffusion reaction equations, as the exact results are known to produce
good results for both linear and non-linear advection diffusion reaction equations.