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Volume 11, Issue 2
Parameter-Uniform Convergence for a Finite Difference Method for a Singularly Perturbed Linear Reaction-Diffusion System with Discontinuous Source Terms

M. paramasivam, J. J. H. Miller & S. Valarmathi

Int. J. Numer. Anal. Mod., 11 (2014), pp. 385-399.

Published online: 2014-11

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  • Abstract

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with discontinuous source terms is considered. A small positive parameter multiplies the leading term of each equation. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlapping boundary and interior layers. A numerical method is constructed that uses a classical finite difference scheme on a piecewise uniform Shishkin mesh. It is proved that the numerical approximations obtained by this method are essentially first order convergent uniformly with respect to all of the perturbation parameters. Numerical illustrations are presented in support of the theory.

  • Keywords

Singular perturbation problems, system of differential equations, reaction-diffusion equations, discontinuous source terms, overlapping boundary and interior layers, classical finite difference scheme, Shishkin mesh, parameter-uniform convergence.

  • AMS Subject Headings

65L10, 65L12, 65L20, 65L70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-11-385, author = {paramasivam , M.Miller , J. J. H. and Valarmathi , S.}, title = {Parameter-Uniform Convergence for a Finite Difference Method for a Singularly Perturbed Linear Reaction-Diffusion System with Discontinuous Source Terms}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2014}, volume = {11}, number = {2}, pages = {385--399}, abstract = {

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with discontinuous source terms is considered. A small positive parameter multiplies the leading term of each equation. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlapping boundary and interior layers. A numerical method is constructed that uses a classical finite difference scheme on a piecewise uniform Shishkin mesh. It is proved that the numerical approximations obtained by this method are essentially first order convergent uniformly with respect to all of the perturbation parameters. Numerical illustrations are presented in support of the theory.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/533.html} }
TY - JOUR T1 - Parameter-Uniform Convergence for a Finite Difference Method for a Singularly Perturbed Linear Reaction-Diffusion System with Discontinuous Source Terms AU - paramasivam , M. AU - Miller , J. J. H. AU - Valarmathi , S. JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 385 EP - 399 PY - 2014 DA - 2014/11 SN - 11 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/533.html KW - Singular perturbation problems, system of differential equations, reaction-diffusion equations, discontinuous source terms, overlapping boundary and interior layers, classical finite difference scheme, Shishkin mesh, parameter-uniform convergence. AB -

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with discontinuous source terms is considered. A small positive parameter multiplies the leading term of each equation. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlapping boundary and interior layers. A numerical method is constructed that uses a classical finite difference scheme on a piecewise uniform Shishkin mesh. It is proved that the numerical approximations obtained by this method are essentially first order convergent uniformly with respect to all of the perturbation parameters. Numerical illustrations are presented in support of the theory.

paramasivam , M.Miller , J. J. H. and Valarmathi , S.. (2014). Parameter-Uniform Convergence for a Finite Difference Method for a Singularly Perturbed Linear Reaction-Diffusion System with Discontinuous Source Terms. International Journal of Numerical Analysis and Modeling. 11 (2). 385-399. doi:
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