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 l∞ 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 l∞ 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
- Introduction 1
- Formulation of the problem . . . . . . . 1
- Numerical Schemes 3
- Finite difference approximations of (1.1) . . . . . . 3
2.2 Some preliminaries . . . . . 7
2.3 Existence and Uniqueness of solution . .. . . . 9
- Consistency-Stability 11
- Consistency Analysis……………….. 11
- Stability analysis……………………… 16
- Convergence analysis 21
- Convergence of the explicit scheme………… 21
- Convergence of the implicit scheme………………… 24
- Numerical simulations and future works 27
- Numerical examples for (2.9)…………….. 28
- Numerical examples for (2.10)…….. 33
Concluding remarks and Future works 38
INTRODUCTION
This work falls within the general areas of numerical methods for partial differ- ential 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.
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J.B. Munyakazi and K. Paditar, A fitted numerical method for singularly perturbed parabolic reaction diffusion problems, Computational and Applied Mathematics, Vol 32, No. 3, 509–519, 2013.