Second-order Numerical Scheme for Singularly Perturbed Reaction-Diffusion Robin Problems

In this article, we consider singularly perturbed reaction-diffusion Robin boundary-value problems. To solve these problems we construct a numerical method which involves both the cubic spline and classical finite difference schemes. The proposed scheme is applied on a piece-wise uniform Shishkin mesh. Truncation error is obtained, and the stability of the method is analyzed. Also, parameter-uniform errors estimates are derived. Three test problems are experimented.

[1]  Third-order variable-mesh cubic spline methods for singularly-perturbed boundary-value problems , 1993 .

[2]  Jesús Vigo-Aguiar,et al.  A Parallel Boundary Value Technique for Singularly Perturbed Two-Point Boundary Value Problems , 2004, The Journal of Supercomputing.

[3]  S. Natesan,et al.  Booster Method for Singularly-Perturbed One-Dimensional Reaction-Diffusion Neumann Problems , 2000 .

[4]  Jesús Vigo-Aguiar,et al.  A numerical algorithm for singular perturbation problems exhibiting weak boundary layers , 2003 .

[5]  Wil H. A. Schilders,et al.  Uniform Numerical Methods for Problems with Initial and Boundary Layers , 1980 .

[6]  R. E. Jr. O'Malley.  Nonlinear singular perturbation phenomena: Theory and application , 1986 .

[7]  Lubin G. Vulkov,et al.  High order ε-uniform methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and singular sources , 2004 .

[8]  R. Chine,et al.  A HYBRID ASYMPTOTIC-FINITE ELEMENT METHOD FOR STIFF TWO-POINT BOUNDARY VALUE PROBLEMS* , 1983 .

[9]  Lutz Tobiska,et al.  Numerical Methods for Singularly Perturbed Differential Equations , 1996 .

[10]  JohnM . Miller,et al.  Robust Computational Techniques for Boundary Layers , 2000 .

[11]  Mohan K. Kadalbajoo,et al.  Variable-mesh difference scheme for singularly-perturbed boundary-value problems using splines , 1996 .

[12]  Srinivasan Natesan,et al.  A computational method for self-adjoint singular perturbation problems using quintic spline , 2005 .

[13]  F. Howes,et al.  Nonlinear Singular Perturbation Phenomena: Theory and Applications , 1984 .

[14]  Jesús Vigo-Aguiar,et al.  An efficient numerical method for singular perturbation problems , 2006 .

[15]  J. J. Miller,et al.  Fitted Numerical Methods for Singular Perturbation Problems , 1996 .