## ABSTRACT

A naive finite difference approximations for singularly

perturbed parabolic reaction-diffusion problems

In this thesis, we treated a Standard Finite Difference Scheme for a singularly

perturbed parabolic reaction-diffusion equation. We proved that the Standard

Finite Difference Scheme is not a robust technique for solving such problems

with singularities. First we discretized the continuous problem in time using the

forward Euler method. We proved that the discrete problem satisfied a stability

property in the l1 norm and l2 norm which is not uniform with respect to

the perturbation parameter, as the solution is unbounded when the perturbation

parameter goes to zero. Error analysis also showed that the solution of the

SFDS is not uniformly convergent in the discrete l1 norm with respect to

the perturbation parameter, (i.e., the convergence is very poor as the parameter

becomes very small). Finally we presented numerical results that confirmed our

theoretical findings.

** **

## TABLE OF CONTENTS

1 Introduction 1

1.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . 1

2 Numerical Schemes 3

2.1 Finite difference approximations of (1.1) . . . . . . . . . . . . . . 3

2.2 Some preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Existence and Uniqueness of solution . . . . . . . . . . . . . . . . 9

3 Consistency-Stability 11

3.1 Consistency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Convergence analysis 21

4.1 Convergence of the explicit scheme . . . . . . . . . . . . . . . . . 21

4.2 Convergence of the implicit scheme . . . . . . . . . . . . . . . . . 24

5 Numerical simulations and future works 27

5.1 Numerical examples for (2.9) . . . . . . . . . . . . . . . . . . . . . 28

5.2 Numerical examples for (2.10) . . . . . . . . . . . . . . . . . . . . 33

5.3 Concluding remarks and Future works . . . . . . . . . . . . . . . 38

## CHAPTER ONE

ntroduction

This work falls within the general areas of numerical methods for partial differential

equations (PDE), an area which prominent mathematicians have explored

due to its diverse applications in numerous fields of sciences. This is evident since

most D.Es can not be solved analytically, thus the method gives us useful insights

into the solutions of the D.Es without necessarily solving them analytically.

1.1 Formulation of the problem

Standard Finite Difference Scheme is one of the most frequently used methods

for solving differential equations numerically. To this end, we study a naive finite

difference approximations for singularly perturbed parabolic reaction-diffusion

problems. The governing equation of the problem is given by:

8><

>:

ut “uxx + b(x; t)u = f(x; t) (x; t) 2 Q =

(0; T]

u(x; 0) = 0 x 2

= [0; 1]

u(0; t) = u(1; t) = 0 t 2 (0; T];

(1.1)

where b(x; t) > 0 for all (x; t) 2

[0; T], ” is the positive perturbation

parameter and f(x; t) is the external force. The diffusion term is uxx, while the reaction

term is b(x; t)u. The problem (1.1) is generally called singularly perturbed

partial differential equation because of the small parameter ” in front of the second

order derivative term in space uxx. Thus (1.1) is one in which a small positive per

turbation parameter ” is multiplied to the highest derivative term in the equa

tion of the problem: Problems of these nature are well known in the literature of

Nnakwe Monday Ogudu 1

partial differential equations as they constitute an element of interest in the area

of population dynamics and chemical reactors, and their numerical analysis is

hard because of the presence of singularity when ” goes to 0: The existence and

uniqueness result of (1.1) is well developed (see [4]). The objective of this thesis

is to show that a naive numerical methods for (1.1) fails when ” goes to 0: To

have an insight into the study, if one takes the stationary problem (as in [5])

8><

>:

” 00 + 0 = 1

2 0 < x < 1 0 < << 1;

(0) = 0;

(1) = 1:

(1.2)

The exact solution of (1.2) is

(x; ) =

1 expx

2(1 exp1

)

+

x

2

:

Thus the solution as

lim

!0

(x; ) =

1 + x

2

= 0(x)

does not live in C2[0; 1] since 0(x) does not satisfy the boundary condition at

x = 0. So we infer that the solution is badly behaved.

In Chapter 2, we introduced the notion of the classical SFD approximation accompanied

with some basic definitions and results. Then we formulated the classical

SFD schemes for (1.1), an elegant proof of the existence and uniqueness of the

solution of the discrete problem was presented.

In Chapter 3, we investigated the consistency and stability of the schemes of the

continuous problem (1.1). It turned out that the stability was not uniform with

respect to the perturbation parameters “:

In Chapter 4, we studied the convergence of the schemes to our continuous problem

(1.1). It turned out that the convergence was very poor as ” goes to zero.

Basically this is why the classical SFDM failed to approximate (1.1), it had no

control over ” and it found itself in damaging position.

In Chapter 5, computer programs were written and simulated for the several

cases of interest and the numerical investigations corroborated with our theoretical

findings.

2

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