Volume 11, Issue 2
On a Numerical Technique to Study Difference Schemes for Singularly Perturbed Parabolic Reaction-diffusion Equations

G. Shishkin, L. Shishkina, J. Gracia & C. Clavero

Int. J. Numer. Anal. Mod., 11 (2014), pp. 412-426

Published online: 2014-11

Preview Purchase PDF 3 2741
Export citation
  • Abstract

A new technique to study special difference schemes numerically for a Dirichlet problem on a rectangular domain (in x, t) is considered for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ε ; ε ∈ (0, 1]. A well known difference scheme on a piecewise-uniform grid is used to solve the problem. Such a scheme converges ε-uniformly in the maximum norm at the rate O(N^{-2} ln^2 N + N^{-1}_0) as N, N_0 → ∞, where N + 1 and N_0 + 1 are the numbers of nodes in the spatial and time meshes, respectively; for ε ≥ m ln^{-1} N the scheme converges at the rate O(N^{-2} + N^{-1}_0). In this paper we elaborate a new approach based on the consideration of regularized errors in discrete solutions, i.e., total errors (with respect to both variables x and t), and also fractional errors (in x and in t) generated in the approximation of differential derivatives by grid derivatives. The regularized total errors agree well with known theoretical estimates for actual errors and their convergence rate orders. It is also shown that a “standard” approach based on the “fine grid technique” turns out inefficient for numerical study of difference schemes because this technique brings to large errors already when estimating the total actual error.

  • Keywords

parabolic reaction-diffusion equation perturbation parameter boundary layer difference scheme piecewise-uniform grids &#949-uniform convergence numerical experiments total error fractional errors regularized errors

  • AMS Subject Headings

35B25 35B45 65M

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{IJNAM-11-412, author = {G. Shishkin, L. Shishkina, J. Gracia and C. Clavero}, title = {On a Numerical Technique to Study Difference Schemes for Singularly Perturbed Parabolic Reaction-diffusion Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2014}, volume = {11}, number = {2}, pages = {412--426}, abstract = {A new technique to study special difference schemes numerically for a Dirichlet problem on a rectangular domain (in x, t) is considered for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ε ; ε ∈ (0, 1]. A well known difference scheme on a piecewise-uniform grid is used to solve the problem. Such a scheme converges ε-uniformly in the maximum norm at the rate O(N^{-2} ln^2 N + N^{-1}_0) as N, N_0 → ∞, where N + 1 and N_0 + 1 are the numbers of nodes in the spatial and time meshes, respectively; for ε ≥ m ln^{-1} N the scheme converges at the rate O(N^{-2} + N^{-1}_0). In this paper we elaborate a new approach based on the consideration of regularized errors in discrete solutions, i.e., total errors (with respect to both variables x and t), and also fractional errors (in x and in t) generated in the approximation of differential derivatives by grid derivatives. The regularized total errors agree well with known theoretical estimates for actual errors and their convergence rate orders. It is also shown that a “standard” approach based on the “fine grid technique” turns out inefficient for numerical study of difference schemes because this technique brings to large errors already when estimating the total actual error.}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/535.html} }
TY - JOUR T1 - On a Numerical Technique to Study Difference Schemes for Singularly Perturbed Parabolic Reaction-diffusion Equations AU - G. Shishkin, L. Shishkina, J. Gracia & C. Clavero JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 412 EP - 426 PY - 2014 DA - 2014/11 SN - 11 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/535.html KW - parabolic reaction-diffusion equation KW - perturbation parameter KW - boundary layer KW - difference scheme KW - piecewise-uniform grids KW - ε-uniform convergence KW - numerical experiments KW - total error KW - fractional errors KW - regularized errors AB - A new technique to study special difference schemes numerically for a Dirichlet problem on a rectangular domain (in x, t) is considered for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ε ; ε ∈ (0, 1]. A well known difference scheme on a piecewise-uniform grid is used to solve the problem. Such a scheme converges ε-uniformly in the maximum norm at the rate O(N^{-2} ln^2 N + N^{-1}_0) as N, N_0 → ∞, where N + 1 and N_0 + 1 are the numbers of nodes in the spatial and time meshes, respectively; for ε ≥ m ln^{-1} N the scheme converges at the rate O(N^{-2} + N^{-1}_0). In this paper we elaborate a new approach based on the consideration of regularized errors in discrete solutions, i.e., total errors (with respect to both variables x and t), and also fractional errors (in x and in t) generated in the approximation of differential derivatives by grid derivatives. The regularized total errors agree well with known theoretical estimates for actual errors and their convergence rate orders. It is also shown that a “standard” approach based on the “fine grid technique” turns out inefficient for numerical study of difference schemes because this technique brings to large errors already when estimating the total actual error.
G. Shishkin, L. Shishkina, J. Gracia & C. Clavero. (1970). On a Numerical Technique to Study Difference Schemes for Singularly Perturbed Parabolic Reaction-diffusion Equations. International Journal of Numerical Analysis and Modeling. 11 (2). 412-426. doi:
Copy to clipboard
The citation has been copied to your clipboard