Analysis of an Il’in Scheme for a System of Singularly Perturbed Convection-Diffusion Equations

Mohammad Ghorbanzadeh,Asghar Kerayechian

doi:10.4236/am.2011.27116

Abstract

In this paper, a numerical solution for a system of singularly perturbed convection-diffusion equations is studied. The system is discretized by the Il’in scheme on a uniform mesh. It is proved that the numerical scheme has first order accuracy, which is uniform with respect to the perturbation parameters. We show that the condition number of the discrete linear system obtained from applying the Il’in scheme for a system of singularly perturbed convection-diffusion equations is O(N) and the relevant coefficient matrix is well conditioned in comparison with the matrices obtained from applying upwind finite difference schemes on this problem. Numerical results confirm the theory of the method.

Highlights

We show that the condition number of the discrete linear system obtained from applying the Il’in scheme for a system of singularly perturbed convection-diffusion equations is O(N) and the relevant coefficient matrix is well conditioned in comparison with the matrices obtained from applying upwind finite difference schemes on this problem
Linss [8] considered an upwind finite difference scheme on special layer adapted Shishkin and Bakhvalov meshes. He showed that the error in the discrete maximum norm is bounded by CN 1 ln N and CN 1 for Shishkin and Bakhvalov meshes respectively, where
Roos in [9] showed that the condition number of the discrete linear system associated with the upwind schemes on Shishkin meshes for a single equation is

Summary

Introduction

T. Linss [8] considered an upwind finite difference scheme on special layer adapted Shishkin and Bakhvalov meshes. Linss [8] considered an upwind finite difference scheme on special layer adapted Shishkin and Bakhvalov meshes He showed that the error in the discrete maximum norm is bounded by CN 1 ln N and CN 1 for Shishkin and Bakhvalov meshes respectively, where. Roos in [9] showed that the condition number of the discrete linear system associated with the upwind schemes on Shishkin meshes for a single equation is. The condition number of the discrete linear system associated with the Il’in scheme on uniform meshes for single equation is O N , which is better in comparison with the precondition of upwind schemes on Shishkin meshes.

Properties of the Exact Solutions

Discretization

Error Analysis

Numerical Experiments

11. References