Scheme of the Solution Decomposition Method for a Singularly Perturbed Reaction-Diffusion Equation
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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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