Why Classical Finite Difference Approximations Fail For A Singularly Perturbed System Of Convection-Diffusion Equations

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Why Classical Finite Difference Approximations Fail For A Singularly Perturbed System Of Convection-Diffusion Equations.

Abstract

We consider classical Finite Difference Scheme for a system of singularly per- turbed 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.

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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 perturba- tion parameters.

Finally we present numerical results that confirm our theoretical findings.

TABLE OF CONTENTS

• Introduction                                                                   1

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

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

• Numerical Schemes                                                      5
  • Finite difference approximation . . . . . . . . .  5
• Consistency-Stability                                                  13
  • consistency analysis……………………………… 13
  • Stability analysis………………………………….  17
• Convergence                                                                 25
• Numerical simulations and future works              29
  • Numerical examples…………………………….. 29

Conclusion and Future Research                                       40

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.

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REFERENCES

B. Munyakazi, A uniformly convergent nonstandard finite difference for a system of convection diffusion equations. Computational and Applied Math- ematics, Vol 34, No. 3, 1153–1165, 2015.
Lucquin and O. Pironneau, Introduction to Scientific Computing. Wiley, 1998.
Grossmann, H. Gorg Roos, M. Stynes, Numerical Treatment of Partial Differential Equations. Springer Verlag, 2007.
Tveito and R. Winther, Introduction to Partial Differential Equations: A Computational Approach. Springer Verlag, 1998.
G. Roos, M. Stynes and L.Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Convection-Diffusion and Flow Problems), Springer-Verlag Berlin Heidelberg , 1996.