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Volume 10, Issue 1
Second Order Parameter-Uniform Convergence for a Finite Difference Method for a Singularly Perturbed Linear Parabolic System

V. Franklin, M. Paramasivam, J.J.H. Miller & S. Valarmathi

Int. J. Numer. Anal. Mod., 10 (2013), pp. 178-202.

Published online: 2013-10

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  • Abstract

A singularly perturbed linear system of second order partial differential equations of parabolic reaction-diffusion type with given initial and boundary conditions is considered. The diffusion term of each equation is multiplied by a small positive parameter. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that in the maximum norm the numerical approximations obtained with this method are first order convergent in time and essentially second order convergent in the space variable, uniformly with respect to all of the parameters.

  • Keywords

Singular perturbation problems, parabolic problems, boundary layers, uniform convergence, finite difference scheme, Shishkin mesh.

  • AMS Subject Headings

65M06, 65N06, 65N12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-10-178, author = {V. Franklin, M. Paramasivam, J.J.H. Miller and S. Valarmathi}, title = {Second Order Parameter-Uniform Convergence for a Finite Difference Method for a Singularly Perturbed Linear Parabolic System}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {1}, pages = {178--202}, abstract = {

A singularly perturbed linear system of second order partial differential equations of parabolic reaction-diffusion type with given initial and boundary conditions is considered. The diffusion term of each equation is multiplied by a small positive parameter. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that in the maximum norm the numerical approximations obtained with this method are first order convergent in time and essentially second order convergent in the space variable, uniformly with respect to all of the parameters.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/564.html} }
TY - JOUR T1 - Second Order Parameter-Uniform Convergence for a Finite Difference Method for a Singularly Perturbed Linear Parabolic System AU - V. Franklin, M. Paramasivam, J.J.H. Miller & S. Valarmathi JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 178 EP - 202 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/564.html KW - Singular perturbation problems, parabolic problems, boundary layers, uniform convergence, finite difference scheme, Shishkin mesh. AB -

A singularly perturbed linear system of second order partial differential equations of parabolic reaction-diffusion type with given initial and boundary conditions is considered. The diffusion term of each equation is multiplied by a small positive parameter. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that in the maximum norm the numerical approximations obtained with this method are first order convergent in time and essentially second order convergent in the space variable, uniformly with respect to all of the parameters.

V. Franklin, M. Paramasivam, J.J.H. Miller and S. Valarmathi. (2013). Second Order Parameter-Uniform Convergence for a Finite Difference Method for a Singularly Perturbed Linear Parabolic System. International Journal of Numerical Analysis and Modeling. 10 (1). 178-202. doi:
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