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.   

Robust Finite Difference Method for Singularly Perturbed Two-Parameter Parabolic Convection-Diffusion Problems

2020 ◽  
pp. 2050034
Author(s):  
Tesfaye Aga Bullo ◽  
Gemechis File Duressa ◽  
Guy Aymard Degla
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Perturbation Parameter ◽  
Accurate Solution ◽  
Singularly Perturbed ◽  
Parabolic Problems ◽  
Uniform Mesh ◽  
Convection Diffusion ◽  
Diffusion Problems

Robust finite difference method is introduced in order to solve singularly perturbed two parametric parabolic convection-diffusion problems. In order to discretize the solution domain, Micken’s type discretization on a uniform mesh is applied and then followed by the fitted operator approach. The convergence of the method is established and observed to be first-order convergent, but it is accelerated by Richardson extrapolation. To validate the applicability of the proposed method, some numerical examples are considered and observed that the numerical results confirm the agreement of the method with the theoretical results effectively. Furthermore, the method is convergent regardless of perturbation parameter and produces more accurate solution than the standard methods for solving singularly perturbed parabolic problems.

scholarly journals SCHEMES CONVERGENT Ε-UNIFORMLY FOR PARABOLIC SINGULARLY PERTURBED PROBLEMS WITH A DEGENERATING CONVECTIVE TERM AND A DISCONTINUOUS SOURCE

2015 ◽  
Vol 20 (5) ◽  
pp. 641-657 ◽  
Author(s):  
Carmelo Clavero ◽  
Jose Luis Gracia ◽  
Grigorii I. Shishkin ◽  
Lidia P. Shishkina
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Euler Method ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Interior Layer ◽  
Convective Term ◽  
Singularly Perturbed Problems ◽  
First Order

We consider the numerical approximation of a 1D singularly perturbed convection-diffusion problem with a multiply degenerating convective term, for which the order of degeneracy is 2p + 1, p is an integer with p ≥ 1, and such that the convective flux is directed into the domain. The solution exhibits an interior layer at the degeneration point if the source term is also a discontinuous function at this point. We give appropriate bounds for the derivatives of the exact solution of the continuous problem, showing its asymptotic behavior with respect to the perturbation parameter ε, which is the diffusion coefficient. We construct a monotone finite difference scheme combining the implicit Euler method, on a uniform mesh, to discretize in time, and the upwind finite difference scheme, constructed on a piecewise uniform Shishkin mesh condensing in a neighborhood of the interior layer region, to discretize in space. We prove that the method is convergent uniformly with respect to the parameter ε, i.e., ε-uniformly convergent, having first order convergence in time and almost first order in space. Some numerical results corroborating the theoretical results are showed.

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.

scholarly journals A Higher-Order Finite Difference Scheme for Singularly Perturbed Parabolic Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Shifang Tian ◽  
Xiaowei Liu ◽  
Ran An
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Diffusion Problem ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Third Order ◽  
Order Accuracy ◽  
Convection Diffusion ◽  
Backward Euler

In this paper, we deal with a singularly perturbed parabolic convection-diffusion problem. Shishkin mesh and a hybrid third-order finite difference scheme are adopted for the spatial discretization. Uniform mesh and the backward Euler scheme are used for the temporal discretization. Furthermore, a preconditioning approach is also used to ensure uniform convergence. Numerical experiments show that the method is first-order accuracy in time and almost third-order accuracy in space.

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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