Robust higher order finite difference scheme for singularly perturbed turning point problem with two outflow boundary layers

2021 ◽  
Vol 40 (5) ◽  
Author(s):  
Vikas Gupta ◽  
Sanjay K. Sahoo ◽  
Ritesh K. Dubey
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Boundary Layers ◽  
Finite Difference Scheme ◽  
Turning Point ◽  
Higher Order ◽  
Singularly Perturbed ◽  
Point Problem ◽  
Outflow Boundary

scholarly journals Finite difference scheme for singularly perturbed reaction diffusion problem of partial delay differential equation with nonlocal boundary condition

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sekar Elango ◽  
Ayyadurai Tamilselvan ◽  
R. Vadivel ◽  
Nallappan Gunasekaran ◽  
Haitao Zhu ◽  
...  
Keyword(s):  
Boundary Condition ◽  
Finite Difference ◽  
Difference Scheme ◽  
Boundary Layers ◽  
Finite Difference Scheme ◽  
Nonlocal Boundary ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Integral Boundary ◽  
Delay Term

AbstractThis paper investigates singularly perturbed parabolic partial differential equations with delay in space, and the right end plane is an integral boundary condition on a rectangular domain. A small parameter is multiplied in the higher order derivative, which gives boundary layers, and due to the delay term, one more layer occurs on the rectangle domain. A numerical method comprising the standard finite difference scheme on a rectangular piecewise uniform mesh (Shishkin mesh) of $N_{r} \times N_{t}$ N r × N t elements condensing in the boundary layers is suggested, and it is proved to be parameter-uniform. Also, the order of convergence is proved to be almost two in space variable and almost one in time variable. Numerical examples are proposed to validate the theory.

A parameter-uniform finite difference scheme for singularly perturbed parabolic problem with two small parameters

2021 ◽  
Author(s):  
Tesfaye Aga Bullo ◽  
Guy Aymard Degla ◽  
Gemechis File Duressa
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Variable Parameter ◽  
Accurate Solution ◽  
Singularly Perturbed ◽  
Finite Difference Approximation ◽  
Parabolic Problems ◽  
Difference Approximation ◽  
Uniform Mesh

A parameter-uniform finite difference scheme is constructed and analyzed for solving singularly perturbed parabolic problems with two parameters. The solution involves boundary layers at both the left and right ends of the solution domain. A numerical algorithm is formulated based on uniform mesh finite difference approximation for time variable and appropriate piecewise uniform mesh for the spatial variable. Parameter-uniform error bounds are established for both theoretical and experimental results and observed that the scheme is second-order convergent. Furthermore, the present method produces a more accurate solution than some methods existing in the literature.   

THE PARAMETER-ROBUST NUMERICAL METHOD BASED ON DEFECT-CORRECTION TECHNIQUE FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH LAYER BEHAVIOR

2010 ◽  
Vol 07 (04) ◽  
pp. 573-594 ◽  
Author(s):  
JUGAL MOHAPATRA ◽  
SRINIVASAN NATESAN
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Perturbation Parameter ◽  
Correction Method ◽  
Singularly Perturbed ◽  
Defect Correction ◽  
Central Difference ◽  
Correction Technique ◽  
Delay Differential

In this article, we consider a defect-correction method based on finite difference scheme for solving a singularly perturbed delay differential equation. We solve the equation using upwind finite difference scheme on piecewise-uniform Shishkin mesh, then apply the defect-correction technique that combines the stability of the upwind scheme and the higher-order central difference scheme. The method is shown to be convergent uniformly in the perturbation parameter and almost second-order convergence measured in the discrete maximum norm is obtained. Numerical results are presented, which are in agreement with the theoretical findings.

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