Parameter-uniform approximation on equidistributed meshes for singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions

2021 ◽  
Vol 392 ◽  
pp. 125677
Author(s):  
Sunil Kumar ◽  
Sumit ◽  
Higinio Ramos
Keyword(s):  
Boundary Conditions ◽  
Uniform Approximation ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Robin Boundary ◽  
Diffusion Problems

A parameter uniform essentially first-order convergent numerical method for a parabolic system of singularly perturbed differential equations of reaction–diffusion type with initial and Robin boundary conditions

2019 ◽  
Vol 12 (01) ◽  
pp. 1950001 ◽  
Author(s):  
R. Ishwariya ◽  
J. J. H. Miller ◽  
S. Valarmathi
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Method ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Diffusion Type ◽  
Parabolic Boundary ◽  
First Order ◽  
Robin Boundary

In this paper, a class of linear parabolic systems of singularly perturbed second-order differential equations of reaction–diffusion type with initial and Robin boundary conditions is considered. The components of the solution [Formula: see text] of this system are smooth, whereas the components of [Formula: see text] exhibit parabolic boundary layers. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first-order convergent in time and essentially first-order convergent in the space variable in the maximum norm uniformly in the perturbation parameters.

scholarly journals An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Li-Bin Liu ◽  
Ying Liang ◽  
Xiaobing Bao ◽  
Honglin Fang
Keyword(s):  
Boundary Conditions ◽  
Grid Method ◽  
Posteriori Error ◽  
Adaptive Grid ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Euler Formula ◽  
Convection Diffusion ◽  
Backward Euler ◽  
Robin Boundary

AbstractA system of singularly perturbed convection-diffusion equations with Robin boundary conditions is considered on the interval $[0,1]$ [ 0 , 1 ] . It is shown that any solution of such a problem can be expressed to a system of first-order singularly perturbed initial value problem, which is discretized by the backward Euler formula on an arbitrary nonuniform mesh. An a posteriori error estimation in maximum norm is derived to design an adaptive grid generation algorithm. Besides, in order to establish the initial values of the original problems, we construct a nonlinear optimization problem, which is solved by the Nelder–Mead simplex method. Numerical results are given to demonstrate the performance of the presented method.

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