Discrete approximation for a two-parameter singularly perturbed boundary value problem having discontinuity in convection coefficient and source term

Prabha, Thara V.; Chandru, M.; Shanthi, V.; Ramos, Higinio

doi:10.1016/j.cam.2019.03.040

Cited by 14 publications

(8 citation statements)

References 12 publications

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“…This is an improvement of the result of Prabha et al. in [21] . Their method has uniform convergence of order one up to a logarithmic factor in the supremum norm.…”

Section: Introductionsupporting

confidence: 66%

“…The bounds presented in the previous section are not sufficient for the error analysis of the discretization method for the singularly perturbed problems. Thus, to obtain sharp bounds, the solution

is decomposed as in [21] into layers and regular components as

. The regular component

is the solution of

The singular components

and

are the solutions of

and

respectively.…”

Section: Decomposition Of the Solutionmentioning

confidence: 99%

“…Let us decompose the regular part (similar to Prabha et al. [21] ) as

where

and

be the solution of the following problems:

respectively.…”

Section: Decomposition Of the Solutionmentioning

confidence: 99%

“…Prabha et al. in [21] , examined two parameter SPP with discontinuous source and convection-coefficient. They proposed an upwind difference scheme layer adapted Shishkin mesh with a three-point difference scheme at the point of discontinuity.…”

Section: Introductionmentioning

confidence: 99%

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A parameter uniform method for two-parameter singularly perturbed boundary value problems with discontinuous data

Roy¹,

Jha²

2023

MethodsX

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“…This is an improvement of the result of Prabha et al. in [21] . Their method has uniform convergence of order one up to a logarithmic factor in the supremum norm.…”

Section: Introductionsupporting

confidence: 66%

is decomposed as in [21] into layers and regular components as

. The regular component

is the solution of

The singular components

and

are the solutions of

and

respectively.…”

Section: Decomposition Of the Solutionmentioning

confidence: 99%

“…Let us decompose the regular part (similar to Prabha et al. [21] ) as

where

and

be the solution of the following problems:

respectively.…”

Section: Decomposition Of the Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A parameter uniform method for two-parameter singularly perturbed boundary value problems with discontinuous data

Roy¹,

Jha²

2023

MethodsX

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“…The method of averaging is used at the point of discontinuity. Based on the standard upwind finite difference scheme, recently, Prabha et al [29] developed a first-order parameters uniform numerical scheme for the problem (1.2). Recently, by using an adaptive mesh, Chandru et al [30] developed an almost first-order parameters uniform numerical scheme for the problem (1.1).…”

mentioning

confidence: 99%

Uniformly convergent scheme for two‐parameter singularly perturbed problems with non‐smooth data

Kumar

Kumari

2020

Numerical Methods Partial

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A numerical scheme is constructed for the problems in which the diffusion and convection parameters (1 and 2 , respectively) both are small, and the convection and source terms have a jump discontinuity in the domain of consideration. Depending on the magnitude of the ratios 1 ∕ 2 2 , and 2 2 ∕ 1 two different cases have been considered separately. Through rigorous analysis, the theoretical error bounds on the singular and regular components of the solution are obtained separately, which shows that in both cases the method is convergent uniformly irrespective of the size of the parameters 1 , 2. Two test problems are included to validate the theoretical results.

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High‐order convergent methods for singularly perturbed quasilinear problems with integral boundary conditions

Kumar

2020

Math Methods in App Sciences

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In this work, we develop a numerical scheme for a class of singularly perturbed quasilinear problems with integral boundary conditions. The quasilinear equation is discretized using a hybrid scheme, and the composite trapezoidal rule is used to discretize the boundary condition. We construct a general error analysis framework for the discrete scheme. Within this framework, the discrete scheme is shown to be uniformly convergent of on Shishkin meshes and on Bakhvalov meshes. Further, we propose adaptive generation of meshes based on a suitable monitor function and the mesh equidistribution principle. We prove that on these meshes the discrete scheme is uniformly convergent of Our theoretical findings are supported by numerical results obtained through experiments.

show abstract

Discrete approximation for a two-parameter singularly perturbed boundary value problem having discontinuity in convection coefficient and source term

Cited by 14 publications

References 12 publications

A parameter uniform method for two-parameter singularly perturbed boundary value problems with discontinuous data

A parameter uniform method for two-parameter singularly perturbed boundary value problems with discontinuous data

Uniformly convergent scheme for two‐parameter singularly perturbed problems with non‐smooth data

High‐order convergent methods for singularly perturbed quasilinear problems with integral boundary conditions

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