Volume 3, Issue 2
Scheme of the Solution Decomposition Method for a Singularly Perturbed Reaction-Diffusion Equation

GRIGORY SHISHKIN AND LIDIA SHISHKINA

Int. J. Numer. Anal. Mod. B, 3 (2012), pp. 168-184

Published online: 2012-03

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  • Abstract
A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed to construct difference schemes convergent uniformly with respect to a perturbation parameter ε for ε ∈ (0, 1], i.e., ε-uniformly. This approach is based on a decomposition of the discrete solution into the regular and singular components which are solutions of discrete subproblems considered on uniform grids. Using an asymptotic construction technique, a difference scheme of the solution decomposition method is constructed that converges ε-uniformly in the maximum norm at the rate O(N^{-2}ln^2N), where N + 1 is the number of nodes in the grids used; for fixed values of the parameter ε, the scheme converges at the rate O(N^{-2}). For the constructed scheme, approximations of the regular and singular components to the solution and their derivatives up to the second order are studied. A modified scheme of the solution decomposition method is constructed for which the regular component of the solution and its discrete derivatives converge $\varepsilon$-uniformly in the maximum norm at the rate O(N^{-2}) for ε = o(ln^{-1} N).
  • AMS Subject Headings

35B25 35B45 35C20 65L10 65L12 65L20 65L70

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@Article{IJNAMB-3-168, author = {GRIGORY SHISHKIN AND LIDIA SHISHKINA}, title = {Scheme of the Solution Decomposition Method for a Singularly Perturbed Reaction-Diffusion Equation}, journal = {International Journal of Numerical Analysis Modeling Series B}, year = {2012}, volume = {3}, number = {2}, pages = {168--184}, abstract = {A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed to construct difference schemes convergent uniformly with respect to a perturbation parameter ε for ε ∈ (0, 1], i.e., ε-uniformly. This approach is based on a decomposition of the discrete solution into the regular and singular components which are solutions of discrete subproblems considered on uniform grids. Using an asymptotic construction technique, a difference scheme of the solution decomposition method is constructed that converges ε-uniformly in the maximum norm at the rate O(N^{-2}ln^2N), where N + 1 is the number of nodes in the grids used; for fixed values of the parameter ε, the scheme converges at the rate O(N^{-2}). For the constructed scheme, approximations of the regular and singular components to the solution and their derivatives up to the second order are studied. A modified scheme of the solution decomposition method is constructed for which the regular component of the solution and its discrete derivatives converge $\varepsilon$-uniformly in the maximum norm at the rate O(N^{-2}) for ε = o(ln^{-1} N).}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnamb/276.html} }
TY - JOUR T1 - Scheme of the Solution Decomposition Method for a Singularly Perturbed Reaction-Diffusion Equation AU - GRIGORY SHISHKIN AND LIDIA SHISHKINA JO - International Journal of Numerical Analysis Modeling Series B VL - 2 SP - 168 EP - 184 PY - 2012 DA - 2012/03 SN - 3 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnamb/276.html KW - singularly perturbed boundary value problem KW - ordinary differential reaction-diffusion equation KW - discrete solution decomposition KW - asymptotic construction technique KW - difference scheme of a solution decomposition method KW - uniform grids KW - ε-uniform convergence KW - maximum norm KW - approximation of derivatives AB - A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed to construct difference schemes convergent uniformly with respect to a perturbation parameter ε for ε ∈ (0, 1], i.e., ε-uniformly. This approach is based on a decomposition of the discrete solution into the regular and singular components which are solutions of discrete subproblems considered on uniform grids. Using an asymptotic construction technique, a difference scheme of the solution decomposition method is constructed that converges ε-uniformly in the maximum norm at the rate O(N^{-2}ln^2N), where N + 1 is the number of nodes in the grids used; for fixed values of the parameter ε, the scheme converges at the rate O(N^{-2}). For the constructed scheme, approximations of the regular and singular components to the solution and their derivatives up to the second order are studied. A modified scheme of the solution decomposition method is constructed for which the regular component of the solution and its discrete derivatives converge $\varepsilon$-uniformly in the maximum norm at the rate O(N^{-2}) for ε = o(ln^{-1} N).
GRIGORY SHISHKIN AND LIDIA SHISHKINA. (2012). Scheme of the Solution Decomposition Method for a Singularly Perturbed Reaction-Diffusion Equation. International Journal of Numerical Analysis Modeling Series B. 3 (2). 168-184. doi:
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