An efficient numerical method to solve 2D parabolic singularly perturbed coupled systems of convection-diffusion type with multi-parameters on a Bakhvalov–Shishkin mesh

Ram Shiromani; Carmelo Clavero; Ram Shiromani; Carmelo Clavero

doi:10.3934/math.2026076

AIMS Mathematics

2026, Volume 11, Issue 1: 1820-1856. doi: 10.3934/math.2026076

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Research article

An efficient numerical method to solve 2D parabolic singularly perturbed coupled systems of convection-diffusion type with multi-parameters on a Bakhvalov–Shishkin mesh

Ram Shiromani ¹,
Carmelo Clavero ^{2
,
,}

1.
Symbiosis Centre for Information Technology, Symbiosis International (Deemed University), Pune, India
2.
Department of Applied Mathematics, IUMA, University of Zaragoza, Zaragoza, Spain

Received: 23 October 2025 Revised: 16 December 2025 Accepted: 08 January 2026 Published: 20 January 2026
MSC : 35B25, 35J40, 35J25, 65N06, 65N15, 65N12, 65N50

This study addresses the efficient solution of a class of 2D parabolic singularly perturbed weakly coupled systems of convection-diffusion type. In the model problem, small positive parameters appear in both the diffusion and the convection terms. We assume that the diffusion parameters can be distinct, but the convection parameter remains the same for both equations. Then, for sufficiently small values of the parameters, overlapping boundary layers appear on the boundary of the spatial domain. To solve the problem, a numerical method is employed that combines the implicit Euler scheme, defined on a uniform mesh, with the upwind scheme for spatial discretization. Then, if the spatial discretization is carried out on an adequate nonuniform Bakhvalov–Shishkin (BS) mesh, the fully discrete scheme attains uniform convergence, with respect to all perturbation parameters; moreover, it has first-order accuracy in both temporal and spatial variables. Note that the construction of the BS mesh depends on the value and the ratio between the diffusion and the convection parameters, and special generating functions are needed to construct them. Numerical experiments illustrating the performance of the algorithm for some test problems are showed, which corroborate the uniform convergence of the method in agreement with the theoretical results.
- 2D parabolic weakly coupled systems,
- diffusion and convection parameters,
- Bakhvalov–Shishkin meshes,
- uniform convergence
Citation: Ram Shiromani, Carmelo Clavero. An efficient numerical method to solve 2D parabolic singularly perturbed coupled systems of convection-diffusion type with multi-parameters on a Bakhvalov–Shishkin mesh[J]. AIMS Mathematics, 2026, 11(1): 1820-1856. doi: 10.3934/math.2026076

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Abstract

This study addresses the efficient solution of a class of 2D parabolic singularly perturbed weakly coupled systems of convection-diffusion type. In the model problem, small positive parameters appear in both the diffusion and the convection terms. We assume that the diffusion parameters can be distinct, but the convection parameter remains the same for both equations. Then, for sufficiently small values of the parameters, overlapping boundary layers appear on the boundary of the spatial domain. To solve the problem, a numerical method is employed that combines the implicit Euler scheme, defined on a uniform mesh, with the upwind scheme for spatial discretization. Then, if the spatial discretization is carried out on an adequate nonuniform Bakhvalov–Shishkin (BS) mesh, the fully discrete scheme attains uniform convergence, with respect to all perturbation parameters; moreover, it has first-order accuracy in both temporal and spatial variables. Note that the construction of the BS mesh depends on the value and the ratio between the diffusion and the convection parameters, and special generating functions are needed to construct them. Numerical experiments illustrating the performance of the algorithm for some test problems are showed, which corroborate the uniform convergence of the method in agreement with the theoretical results.

References

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