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.
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