A Naive Finite Difference Approximations for Singularly Perturbed Parabolic Reaction-diffusion Problems

A Naive Finite Difference Approximations for Singularly Perturbed Parabolic Reaction-diffusion Problems

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

Introduction

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 perturbation parameter ” is multiplied to the highest derivative term in the equation 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 ? exp?x 2(1 ? exp?1 ) + 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.