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

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Download Why Classical Finite Difference Approximations Fail For A Singularly Perturbed System Of Convection-Diffusion Equations. Mathematics students who are writing their projects can get this material to aid their research work.

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.

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.

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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