A non-uniform difference scheme for solving singularly perturbed 1D-parabolic reaction–convection–diffusion systems with two small parameters and discontinuous source terms

Aarthika, K.; Shanthi, V.; Ramos, Higinio

doi:10.1007/s10910-019-01094-1

Cited by 11 publications

(2 citation statements)

References 19 publications

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“…In the literature, several parameter-robust numerical methods (based on the fitted mesh methods) are developed for analyzing linear SPDEs with discontinuous data. To cite a few, for significant research contributions toward linear SPDEs mostly with discontinuous convection coefficient, one can recall the articles [2,[22][23][24][25][26][27] where solution of SPDE possesses strong interior layers and the articles [28][29][30][31] where solution of SPDE generates both boundary and strong interior layers. In this regard, we cite the recent research finding in Shiromani et al [32] for 2D elliptic singularly perturbed convection-diffusion problems with discontinuous convection and source terms, which give rise to strong interior layer phenomena.…”

Section: Introductionmentioning

confidence: 99%

Parameter‐robust higher‐order time‐accurate computational method for singularly perturbed time‐dependent semilinear convection‐diffusion PDEs with discontinuous data

Yadav,

Mukherjee

2024

Math Methods in App Sciences

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This article is concerned with a class of singularly perturbed semilinear parabolic convection‐diffusion partial differential equations (PDEs) with discontinuous source function. Solutions of these PDEs usually exhibit a weak interior layer at one side of the point of discontinuity along with a boundary layer at one side of the spatial domain. We begin our study by proving existence of the analytical solution of the considered nonlinear PDE by means of the upper and lower solutions approach; and the ‐uniform stability of the analytical solution is established by using the comparison principle for the continuous nonlinear operator. In order to realize the asymptotic behavior of the analytical solution, we derive a priori bounds of the solution derivatives via decomposition of the solution into the smooth and the layer components. For an efficient numerical solution of the nonlinear PDE, the time‐derivative is approximated by the Crank–Nicolson method on an equidistant mesh, and we approximate the spatial derivative by a finite difference scheme on a suitable layer‐adapted mesh. We establish the comparison principle for the nonlinear difference operator to prove the ‐uniform stability of the discrete solution and further construct a suitable decomposition of the discrete solution for pursuing the convergence analysis. The computational method is proven to be parameter‐robust with second‐order time accuracy in the discrete supremum norm. The theoretical estimate is finally verified by the numerical experiments.

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Section: Introductionmentioning

confidence: 99%

Parameter‐robust higher‐order time‐accurate computational method for singularly perturbed time‐dependent semilinear convection‐diffusion PDEs with discontinuous data

Yadav,

Mukherjee

2024

Math Methods in App Sciences

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“…Here, we are considering the most common and challenging singularly perturbed partial differential equations (SPPDEs) in the literature, that is of the 'convection-diffusion' type. They appear in a wide range of scientific and engineering fields, and many researchers have focused on these problems (see previous studies [1][2][3][4][5][6] ).…”

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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A Quadratic B-Spline Collocation Method for a Singularly Perturbed Semilinear Reaction–Diffusion Problem with Discontinuous Source Term

Cen

Huang

2023

Mediterr. J. Math.

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A non-uniform difference scheme for solving singularly perturbed 1D-parabolic reaction–convection–diffusion systems with two small parameters and discontinuous source terms

Cited by 11 publications

References 19 publications

Parameter‐robust higher‐order time‐accurate computational method for singularly perturbed time‐dependent semilinear convection‐diffusion PDEs with discontinuous data

Parameter‐robust higher‐order time‐accurate computational method for singularly perturbed time‐dependent semilinear convection‐diffusion PDEs with discontinuous data

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 Quadratic B-Spline Collocation Method for a Singularly Perturbed Semilinear Reaction–Diffusion Problem with Discontinuous Source Term

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