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

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.

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.

KEYWORDS

Convection-Diffusion, Il’in Scheme, Uniform Convergence, Singular Perturbation, Condition Number

Convection-Diffusion, Il’in Scheme, Uniform Convergence, Singular Perturbation, Condition Number

Cite this paper

nullM. Ghorbanzadeh and A. Kerayechian, "Analysis of an Il’in Scheme for a System of Singularly Perturbed Convection-Diffusion Equations,"*Applied Mathematics*, Vol. 2 No. 7, 2011, pp. 866-873. doi: 10.4236/am.2011.27116.

nullM. Ghorbanzadeh and A. Kerayechian, "Analysis of an Il’in Scheme for a System of Singularly Perturbed Convection-Diffusion Equations,"

References

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[6] G. M. Amiraliyev, “The Convergence of a Finite Difference Method on Layer-Adapted Mesh for a Singularly Perturbed System,” Applied Mathematics and Computation, Vol. 162 No. 3, 2005, pp. 1023-1024. doi:10.1016/j.amc.2004.01.015

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[8] T. Lin?, “Analysis of an Upwind Finite-Difference Scheme for a System of Coupled Singularly Perturbed Convection-Diffusion Equations,” Computing, Vol. 79, No. 1, 2007, pp. 23-32. doi:10.1007/s00607-006-0215-x

[9] H. G. Roos, “A Note on the Conditioning of Upwind Schemes on Shishkin Meshes,” IMA Journal of Numerical Analysis, Vol. 16, No. 4, 1996, pp. 529-538. doi:10.1093/imanum/16.4.529

[10] A. M. Il’in, “A Difference Scheme for a Differential Equation with a Small Parameter Affecting the Highest Derivative,” in Russian, Matematicheskie Zametki, Vol. 6, 1969, pp. 237-248.

[11] V. B. Andreev, “The Green Function and A Priori Estimates of Solution of Monotone Three Point Singularly Perturbed Finite-Difference Schemes,” Differrence Equations, Vol. 37, No. 7, 2001, pp. 923-933. doi:10.1023/A:1011949419389

[12] R. B. Kellogg and A. Tsan, “Analysis of Some Difference Approximations for a Singular Perturbation Problem without Turning Points,” Mathematics of Computation, Vol. 32, 1978, pp. 1025-1039. doi:10.1090/S0025-5718-1978-0483484-9

[13] O. Axelsson and L.Kolotilina, “Monotonicity and Discretization Error Estimates,” SIAM Journal on Numerical Analysis, Vol. 27, No. 6, 1990, pp. 1591-1611. doi:10.1090/S0025-5718-1978-0483484-9

[14] H. G. Roos, M. Stynes and L. Tobiska, “Robust Methods for Singularly Perturbed Differential Equations,” 2nd Edition, Springer Series in Computational Mathematics, Springer, Berlin, 2008.

[15] T. Lin?, “Analysis of a System of Singularly Perturbed Covection-Diffusion Equations with Strong Coupling,” SIAM Journal on Numerical Analysis, Vol. 47, No. 3, 2009, pp. 1847-1862.doi:10.1137/070683970

[1] T. Linss and N. Madden, “Accurate Solution of a System of Coupled Singularly Perturbed Reaction-Diffusion Equations,” Computing, Vol. 73, No. 2, 2004, pp. 121-133. doi:10.1007/s00607-004-0065-3

[2] N. Madden and M. Stynes, “A Uniformly Convergent Numerical Method for a Coupled System of Two Singularly Perturbed Linear Reaction-Diffusion Problems,” IMA Journal of Numerical Analysis, Vol. 23, No. 4, 2003, pp. 627-644. doi:10.1093/imanum/23.4.627

[3] J. L. Gracia and F. J. Lisbona, “A Uniformly Convergent Scheme for a System of Reaction-Diffusion Equations,” Journal of Computational and Applied Mathematics, Vol. 206, No. 1, 2007, pp. 1-16. doi:10.1016/j.cam.2006.06.005

[4] S. Bellew and E. O’Riordan, “A Parameter-Robust Numerical Method for a System of Two Singularly Perturbed Convection-Diffusion Equations,” Applied Numerical Mathematics, Vol. 51, No. 2-3, 2004, pp. 171-186. doi:10.1016/j.apnum.2004.05.006

[5] Z. Cen, “Parameter-Uniform Finite Difference Scheme for a System of Coupled Singularly Perturbed Convection-Diffusion Equations,” International Journal of Com- puter Mathematics, Vol. 82, No. 2, 2005, pp. 177-192. doi:10.1080/0020716042000301798

[6] G. M. Amiraliyev, “The Convergence of a Finite Difference Method on Layer-Adapted Mesh for a Singularly Perturbed System,” Applied Mathematics and Computation, Vol. 162 No. 3, 2005, pp. 1023-1024. doi:10.1016/j.amc.2004.01.015

[7] V. B. Andreev and N. Kopteva, “On the Convergence, Uniform with Respect to a Small Parameter of Monotone Three-Point Finite-Difference Approximations,” Journal of Difference Equations, Vol. 34, 1998, pp. 921-929.

[8] T. Lin?, “Analysis of an Upwind Finite-Difference Scheme for a System of Coupled Singularly Perturbed Convection-Diffusion Equations,” Computing, Vol. 79, No. 1, 2007, pp. 23-32. doi:10.1007/s00607-006-0215-x

[9] H. G. Roos, “A Note on the Conditioning of Upwind Schemes on Shishkin Meshes,” IMA Journal of Numerical Analysis, Vol. 16, No. 4, 1996, pp. 529-538. doi:10.1093/imanum/16.4.529

[10] A. M. Il’in, “A Difference Scheme for a Differential Equation with a Small Parameter Affecting the Highest Derivative,” in Russian, Matematicheskie Zametki, Vol. 6, 1969, pp. 237-248.

[11] V. B. Andreev, “The Green Function and A Priori Estimates of Solution of Monotone Three Point Singularly Perturbed Finite-Difference Schemes,” Differrence Equations, Vol. 37, No. 7, 2001, pp. 923-933. doi:10.1023/A:1011949419389

[12] R. B. Kellogg and A. Tsan, “Analysis of Some Difference Approximations for a Singular Perturbation Problem without Turning Points,” Mathematics of Computation, Vol. 32, 1978, pp. 1025-1039. doi:10.1090/S0025-5718-1978-0483484-9

[13] O. Axelsson and L.Kolotilina, “Monotonicity and Discretization Error Estimates,” SIAM Journal on Numerical Analysis, Vol. 27, No. 6, 1990, pp. 1591-1611. doi:10.1090/S0025-5718-1978-0483484-9

[14] H. G. Roos, M. Stynes and L. Tobiska, “Robust Methods for Singularly Perturbed Differential Equations,” 2nd Edition, Springer Series in Computational Mathematics, Springer, Berlin, 2008.

[15] T. Lin?, “Analysis of a System of Singularly Perturbed Covection-Diffusion Equations with Strong Coupling,” SIAM Journal on Numerical Analysis, Vol. 47, No. 3, 2009, pp. 1847-1862.doi:10.1137/070683970