**ABSTRACT**

Why Classical Finite Difference Approximations fail for a singularly

perturbed system of convection-diffusion equations

Msc candidate: Aroh Innocent Tagbo

We consider classical Finite Difference Scheme for a system of singularly perturbed

convection-diffusion equations coupled in their reactive terms, we prove

that the classical SFD scheme is not a robust technique for solving such problem

with singularities. First we prove that the discrete operator satisfies a stability

property in the l2-norm which is not uniform with respect to the perturbation

parameters, as the solution blows up when the perturbation parameters goes to

zero. An error analysis also shows that the solution of the SFD is not uniformly

convergent in the discrete l2-norm with respect to the perturbation parameters,

i.e., the convergence is very poor for a sufficiently small choice of the perturbation

parameters. Finally we present numerical results that confirm our theoretical

findings.

**TABLE OF CONTENTS**

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . 2

2 Numerical Schemes 5

2.1 Finite difference approximation . . . . . . . . . . . . . . . . . . . 5

3 Consistency-Stability 13

3.1 consistency analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Convergence 25

5 Numerical simulations and future works 29

5.1 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 Conclusion and Future Research . . . . . . . . . . . . . . . . . . . 40

**CHAPTER ONE**

Introduction

The contents of this thesis fall within the general area of numerical methods for

PDE, an area which has attracted the attention of prominent mathematicians

due to its diverse applications in numerous fields of sciences

1.1 Motivation

Imagine a river – a river flowing strongly and smoothly, liquid pollution pours into

the water at a certain point, which shape does the pollution stain form on the

surface of the river? Two physical processes operate here: the pollution diffuses

slowly through the water, but the dominant mechanism is the swift movement of

the river which rapidly convects the pollution along a one – dimensional curve on

the surface; diffusion gradually spreads that curve. When convection and diffusion

are both present in a linear differential equation and convection dominates, we

have a convection – diffusion problem. The simplest mathematical model of a

convection – diffusion problem is a two point – point boundary value problem of

the form,

(

”u00(x) + a(x)u0(x) + b(x)u(x) = f(x) 0 < x < 1

u(0) = u(1) = 0;

(1.1)

where ” is a small positive parameter and a(x); b(x); f(x) are some given functions.

The term u00 corresponds to the diffusion and its coefficient ” is small, while the

expression u0 represents convection. Finally u and f(x) play the roles of a source

and driving term respectively.

Aroh Innocent Tagbo 1

Now, having known that the solutions of ODE’s lives in C[a; b], consider the

problem

”u

00

(x) + u0(x) = 1 for 0 < x < 1 : : : : (1.2)

with u(0) = u(1) = 0 and 0 < ” < 1.

suppose that we formally set ” = 0; here we get

(

u0(x) = 1 for 0 < x < 1 : : :

u(0) = u(1) = 0:

(1.3)

The problem (1.3) has no solution in C[0; 1] so we infer than when ” is near

zero the solution of (1.3) is badly behaved. Problems like (1.3) are differential

equations that depend on small positive parameter ” and whose solutions (or

their derivatives ) approach a discontinuous limit as ” approaches zero. We say

that such problems are singularly perturbed where we regard ” as a perturbation

parameter. In more technical terms , one cannot represent the solution of a singularly

perturbed differential equation as an asymptotic expansion in the powers

of “. Moreover not every differential equation be it ODE or PDE can be solved

analytically and singular Perturbations arise in several branches of engineering

and applied mathematics, including fluid dynamics, so in investigating numerical

skills for tackling such problems leads to the main objective of this thesis.

1.2 Formulation of the problem

Classical Finite Difference Scheme is one of the most frequently used method for

numerical solution for both ordinary and partial differential equation. But on the

contrary, in this work we study why classical SFD scheme fails to approximate

a coupled system of singularly perturbed convection-diffusion. The governing

equations of the problem are given by

8><

>:

”uxx a1(x)ux + b11(x)u + b12(x)v = f(x);

vxx a2(x)vx + b21(x)u + b22(x)v = g(x);

u(0) = u(1) = v(0) = v(1) = 0:

(1.4)

where (u; v) is the solution of (1.4) above. In (1.4), we assume that

0 < ” < 1; (1.5)

2

and

ak(x) > 0 ; bkk(x) 0 ; k = 1; 2: (1.6)

The convection-diffusion equation (1.4) are considered as linearised version of

the Navier-Stokes equation, they constitute an element of interest in the area of

fluid dynamics and hydro dynamics. Although the equation (1.4) may not be

applied directly to real applications, it is an important stage in investigation of

many practical applications. There is a lot of work in literature dealing with the

numerical solution of a single equation of (1.4) but systems of equations appear

relatively rare.

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 for (1.4) and showed its consistency with the continuous problem (1.4), we

gave an elegant proof of the existence and uniqueness of the solution of the discrete

operator.

In chapter 3 and chapter 4, stability analysis and error analysis were both investigated

respectively, and both turned out not to be uniform with respect to the

perturbation parameters (“; ). For the stability analysis, the solution blows up

as (“; ) goes to zero, and there will no convergence at all as (“; ) goes to zero.

Basically this is why the classical SFD fail to approximate (1.4), it couldn’t take

care of (“; ) and they found them selves in damaging positions.

In chapter 5, we wrote a computer program and simulate the method for several

cases of interest and the numerical investigations corroborated with our theoretical

findings.

3

4