Parameter-uniform numerical method for a two-dimensional singularly perturbed convection–reaction–diffusion problem with interior and boundary layers

Rao, S. Chandra Sekhara; Chaturvedi, Abhay Kumar

doi:10.1016/j.matcom.2021.03.016

Cited by 12 publications

(9 citation statements)

References 29 publications

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“…Since the solutions obtained here are robust solutions, from theoretical as well as application points of view, the results of this work can be generalized to 2D and 3D problems. In the context of interface problems, the approach developed here can also be used to solve 1D and 2D interface problems (like Rao and Chawla 41 and Rao and Chaturvedi 42 ). These will be the subject of our future research.…”

Section: Discussionmentioning

confidence: 99%

Pointwise error estimates for a system of two singularly perturbed time‐dependent semilinear reaction–diffusion equations

Rao

Chaturvedi

2021

Math Methods in App Sciences

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We present a finite difference method for a system of two singularly perturbed initial-boundary value semilinear reaction-diffusion equations. The highest order derivatives are multiplied by small perturbation parameters of different magnitudes. The problem is discretized using a central difference scheme in space and backward difference scheme in time on a Shishkin mesh. The convergence analysis has been given, and it has been established that the method enjoys almost second-order parameter-uniform convergence in space and first-order in time. Numerical experiments are conducted to demonstrate the efficiency of the method.

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Section: Discussionmentioning

confidence: 99%

Pointwise error estimates for a system of two singularly perturbed time‐dependent semilinear reaction–diffusion equations

Rao

Chaturvedi

2021

Math Methods in App Sciences

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“…It is contingent on the data's smoothness and the fulfillment of numerous compatibility requirements. For the reaction-diffusion problems 44,45 and convection-diffusion problems, 38 the regularity and compatibility criteria have been investigated. Further, Han and Kellogg 42 provide comprehensive knowledge on the importance of differentiability and compatibility requirements.…”

Section: Analytical Properties Of the Solutionmentioning

confidence: 99%

“…(cf., Linß 24 ), and thus, it becomes highly persuading. Rao et al 38 introduced the FDM with Shishkin mesh for singularly perturbed 2-D problems with the interior and boundary layers. Recently, in Shanthi et al, 39 they explained 2-D singularly perturbed reaction-diffusion problems with a non-smooth source term, using the hybrid-difference method with Shishkin mesh.…”

Section: Introduction and The Model Problemmentioning

confidence: 99%

A finite difference method for a singularly perturbed 2‐D elliptic convection‐diffusion PDEs on Shishkin‐type meshes with non‐smooth convection and source terms

Shiromani

Shanthi

Vigo‐Aguiar

2022

Math Methods in App Sciences

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This paper deals with a class of singularly perturbed 2‐D elliptic convection‐diffusion PDEs with non‐smooth convection and source terms. The discontinuity in the convection and source terms portray corner and interior layers in the solution. This type of model problem often appears in modeling various physical phenomena, particularly, in mathematical biology, and thus requires effective numerical techniques for analyzing them computationally. For this purpose, we approximate the considered linear problem by developing an efficient numerical method. The spatial discretization for the numerical method is based on a finite difference scheme. The spatial domain is discretized by appropriate layer‐adapted meshes (S‐mesh and B‐S‐mesh) to accomplish this. The theoretical outcomes are finally supported by extensive numerical experiments, which also include a comparison of the proposed numerical method in terms of the order of accuracy and the computational cost.

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“…The usual task has been to provide means of dealing with the challenges that come with the perturbation parameter and its impact on the solution behavior. While countless successes have been recorded in the case of linear singularly perturbed problems [see for example [2][3][4][5][6][7]], little attention has been paid to the non-linear case.…”

Section: Introductionmentioning

confidence: 99%

A NSFD Discretization of Two-Dimensional Singularly Perturbed Semilinear Convection-Diffusion Problems

Kehinde

Munyakazi

Appadu

2022

Front. Appl. Math. Stat.

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Despite the availability of an abundant literature on singularly perturbed problems, interest toward non-linear problems has been limited. In particular, parameter-uniform methods for singularly perturbed semilinear problems are quasi-non-existent. In this article, we study a two-dimensional semilinear singularly perturbed convection-diffusion problems. Our approach requires linearization of the continuous semilinear problem using the quasilinearization technique. We then discretize the resulting linear problems in the framework of non-standard finite difference methods. A rigorous convergence analysis is conducted showing that the proposed method is first-order parameter-uniform convergent. Finally, two test examples are used to validate the theoretical findings.

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Parameter-uniform numerical method for a two-dimensional singularly perturbed convection–reaction–diffusion problem with interior and boundary layers

Cited by 12 publications

References 29 publications

Pointwise error estimates for a system of two singularly perturbed time‐dependent semilinear reaction–diffusion equations

Pointwise error estimates for a system of two singularly perturbed time‐dependent semilinear reaction–diffusion equations

A finite difference method for a singularly perturbed 2‐D elliptic convection‐diffusion PDEs on Shishkin‐type meshes with non‐smooth convection and source terms

A NSFD Discretization of Two-Dimensional Singularly Perturbed Semilinear Convection-Diffusion Problems

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