An almost second order uniformly convergent method for a two-parameter singularly perturbed problem with a discontinuous convection coefficient and source term

M. Chandru; T. Prabha; V. Shanthi; H. Ramos; M. Chandru; T. Prabha; V. Shanthi; H. Ramos

doi:10.3934/math.20241219

AIMS Mathematics

2024, Volume 9, Issue 9: 24998-25027. doi: 10.3934/math.20241219

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An almost second order uniformly convergent method for a two-parameter singularly perturbed problem with a discontinuous convection coefficient and source term

1.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, Tamilnadu, India
2.
Department of Mathematics, Saranathan College of Engineering, Thiruchirappalli-620012, Tamilnadu, India
3.
Department of Mathematics, National Institute of Technology, Tiruchirappalli-620 015, Tamilnadu, India
4.
Scientific Computing Group, Universidad de Salamanca, Plaza de la Merced 37008, Salamanca, Spain
5.
Escuela Politécnica Superior de Zamora, Universidad de Salamanca, Avda. de Requejo 33, 49029 Zamora, Spain

Received: 01 July 2024 Revised: 07 August 2024 Accepted: 15 August 2024 Published: 27 August 2024
MSC : 65L10

In this paper, we discuss a higher-order convergent numerical method for a two-parameter singularly perturbed differential equation with a discontinuous convection coefficient and a discontinuous source term. The presence of perturbation parameters generates boundary layers, and the discontinuous terms produce interior layers on both sides of the discontinuity. In order to obtain a higher-order convergent solution, a hybrid monotone finite difference scheme is constructed on a piecewise uniform Shishkin mesh, which is adapted inside the boundary and interior layers. On this mesh (including the point of discontinuity), the present method is almost second-order parameter-uniform convergent. The current scheme is compared with the standard upwind scheme, which is used at the point of discontinuity. The numerical experiments based on the proposed scheme show higher-order (almost second-order) accuracy compared to the standard upwind scheme, which provides almost first-order parameter-uniform convergence.
- two-parameter singularly perturbed problem,
- boundary and interior layers,
- hybrid numerical scheme,
- parameter-uniform convergence
Citation: M. Chandru, T. Prabha, V. Shanthi, H. Ramos. An almost second order uniformly convergent method for a two-parameter singularly perturbed problem with a discontinuous convection coefficient and source term[J]. AIMS Mathematics, 2024, 9(9): 24998-25027. doi: 10.3934/math.20241219

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Abstract

In this paper, we discuss a higher-order convergent numerical method for a two-parameter singularly perturbed differential equation with a discontinuous convection coefficient and a discontinuous source term. The presence of perturbation parameters generates boundary layers, and the discontinuous terms produce interior layers on both sides of the discontinuity. In order to obtain a higher-order convergent solution, a hybrid monotone finite difference scheme is constructed on a piecewise uniform Shishkin mesh, which is adapted inside the boundary and interior layers. On this mesh (including the point of discontinuity), the present method is almost second-order parameter-uniform convergent. The current scheme is compared with the standard upwind scheme, which is used at the point of discontinuity. The numerical experiments based on the proposed scheme show higher-order (almost second-order) accuracy compared to the standard upwind scheme, which provides almost first-order parameter-uniform convergence.

References

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