A new stable finite difference scheme and its convergence for time-delayed singularly perturbed parabolic PDEs

Chakravarthy, P. Pramod; Kumar, Kamalesh

doi:10.1007/s40314-020-01170-2

Cited by 23 publications

(9 citation statements)

References 18 publications

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“…In Tables 4 and 5, comparison of the ε-uniform error and ε -uniform rate of convergence of the proposed scheme with results of some recently published papers is given. As we observed, the proposed scheme gives a more accurate result than the results in [16,[18][19][20] and [24]. The proposed scheme has a limitation for solving nonlinear singularly perturbed problems.…”

Section: Numerical Examples Results and Discussionmentioning

confidence: 67%

“…Different authors in [16][17][18][19][20][21][22][23] developed numerical schemes using fitted mesh techniques for treating singularly perturbed parabolic time delay convection-diffusion equations. In [24], Podila and Kumar used nonstandard FDM for treating singularly perturbed parabolic time delay convection-diffusion equations. Their scheme gives linear order of convergence.…”

Section: Introductionmentioning

confidence: 99%

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Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

Woldaregay

Aniley

Duressa

2021

Advances in Mathematical Physics

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This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of ε .

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Section: Numerical Examples Results and Discussionmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

Woldaregay

Aniley

Duressa

2021

Advances in Mathematical Physics

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“…In [8], the scheme is constructed using the Crank-Nicolson method for temporal discretization, and a midpoint upwind finite difference scheme on a fitted piecewise-uniform mesh in spatial discretization is applied. In [15], the scheme is devised using backward Euler's scheme on uniform mesh for temporal discretization and a new stable finite difference scheme on Shishkin mesh for spatial discretization. In [16], the problem is solved using the Crank-Nicolson method in temporal discretization, and in the spatial discretization, an exponentially fitted operator finite difference method on uniform mesh is used.…”

Section: Introductionmentioning

confidence: 99%

Hybrid Fitted Numerical Scheme for Singularly Perturbed Convection-Diffusion Problem with a Small Time Lag

Ayele

Tiruneh

Derese

2023

Abstract and Applied Analysis

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In this article, a singularly perturbed convection-diffusion problem with a small time lag is examined. Because of the appearance of a small perturbation parameter, a boundary layer is observed in the solution of the problem. A hybrid scheme has been constructed, which is a combination of the cubic spline method in the boundary layer region and the midpoint upwind scheme in the outer layer region on a piecewise Shishkin mesh in the spatial direction. For the discretization of the time derivative, the Crank-Nicolson method is used. Error analysis of the proposed method has been performed. We found that the proposed scheme is second-order convergent. Numerical examples are given, and the numerical results are in agreement with the theoretical results. Comparisons are made, and the results of the proposed scheme give more accurate solutions and a higher rate of convergence as compared to some previous findings available in the literature.

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“…The upwind finite difference method on Shishkin mesh is used in [ 6 ]. Podila and Kumar [ 7 ] used a stable finite difference scheme, which works on a uniform and an adaptive mesh. An exponentially fitted scheme is discussed in [ 8 , 9 ].…”

Section: Introductionmentioning

confidence: 99%

Fitted computational method for solving singularly perturbed small time lag problem

et al. 2022

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Objectives An accurate exponentially fitted numerical method is developed to solve the singularly perturbed time lag problem. The solution to the problem exhibits a boundary layer as the perturbation parameter approaches zero. A priori bounds and properties of the continuous solution are discussed. Result The backward-Euler method is applied in the time direction and the higher order finite difference method is employed for the spatial derivative approximation. An exponential fitting factor is induced on the difference scheme for stabilizing the computed solution. Using the comparison principle, the stability of the method is examined and analyzed. It is proved that the method converges uniformly with linear order of convergence. To validate the theoretical findings and analysis, two test examples are given. Comparison is made with the results available in the literature. The proposed method has better accuracy than the schemes in the literature.

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A new stable finite difference scheme and its convergence for time-delayed singularly perturbed parabolic PDEs

Cited by 23 publications

References 18 publications

Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

Hybrid Fitted Numerical Scheme for Singularly Perturbed Convection-Diffusion Problem with a Small Time Lag

Fitted computational method for solving singularly perturbed small time lag problem

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