Fitted Mesh Method for Singularly Perturbed Robin Type Boundary Value Problem with Discontinuous Source Term

Chandru, M.; Shanthi, V.

doi:10.1007/s40819-015-0030-1

Cited by 16 publications

(5 citation statements)

References 20 publications

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“…However, comparatively less research articles are available for analyzing SPDEs possessing both boundary and weak interior layers. In this context, one can refer to the articles [3,33] dealing with linear convection-diffusion BVPs with discontinuous source function. The authors of these articles analyze an almost first-order uniformly convergent numerical scheme.…”

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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“…Shanthi et al [8] obtained a numerical solution of second-order SPBVP with discontinuity term in the right-hand side of the inhomogeneous equation containing two parameters using the fitted mesh method and obtained the error bounds. Researchers have solved the SPBVP with discontinuous source terms by numerical techniques [9][10][11][12] to achieve a better numerical solution. Cen et al [13] obtained fourth-order convergence using a high-order finite difference method having interior layers.…”

Section: Introductionmentioning

confidence: 99%

Hybrid Difference Scheme for Singularly Perturbed Differential Equation with Discontinuous Source Term

Mane,

Lodhi

2024

MMEP

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In this article, we investigate a hybrid difference scheme for finding the numerical solution of a singularly perturbed second-order reaction-diffusion problem with a discontinuous source term. Such types of problems arise in the modeling of semiconductor devices and geophysical fluid dynamics etc. Solutions of these types of problems are difficult to obtain due to the presence of boundary and interior layers. A hybrid difference scheme i.e., cubic spline method and central finite difference approach, are applied on a fine region and coarse region, respectively. Shishkin mesh is utilized to generate the mesh point for the given domain. We use a second-order hybrid difference operator at the point of discontinuity. The solution rapidly changes in the interior layers and boundary layer. Truncation error is studied, and the stability of the method is analyzed. The proposed method is implemented on two problems, and numerical results are compared with the existing method, which shows that the proposed method is efficient for reducing maximum absolute errors and increasing the rate of convergence.

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“…Recently, Chandru and Shanthi [9] presented a fitted mesh method to solve singularly perturbed robin-type boundary value problems with discontinuous source term. But still, there is a room to increase the accuracy and develop its convergence for the developed scheme because the treatment of singular perturbation problem is not trivial distributions and the solution pended on perturbation parameter ε and mesh size h [10].…”

Section: Introductionmentioning

confidence: 99%

“…As far as the researchers' knowledge is concerned, numerical solution of fitted nonpolynomial cubic spline method for the problem under consideration is first being considered. Hence, in the present paper, motivated by the works of [9], we developed a fitted nonpolynomial cubic spline method for singularly perturbed robin-type boundary value problems with discontinuous source term. Therefore, it is important to develop more accurate and convergent numerical method for solving singularly perturbed robin boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

Uniformly Convergent Nonpolynomial Spline Method for Singularly Perturbed Robin-Type Boundary Value Problems with Discontinuous Source Term

Debela¹,

Duressa²

2021

Abstract and Applied Analysis

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In this paper, a singularly perturbed second-order ordinary differential equation with discontinuous source term subject to mixed-type boundary conditions is considered. A fitted nonpolynomial spline method is suggested. The stability and parameter uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε , and mesh size, h . The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε -uniformly convergent for h ≥ ε where the classical numerical methods fail to give good result and it also improves the results of the methods existing in the literature.

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Fitted Mesh Method for Singularly Perturbed Robin Type Boundary Value Problem with Discontinuous Source Term

Cited by 16 publications

References 20 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

Hybrid Difference Scheme for Singularly Perturbed Differential Equation with Discontinuous Source Term

Uniformly Convergent Nonpolynomial Spline Method for Singularly Perturbed Robin-Type Boundary Value Problems with Discontinuous Source Term

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