A numerical approach to approximate the solution of a quasilinear singularly perturbed parabolic convection diffusion problem having a non-smooth source term

Ruby; Vembu Shanthi; Higinio Ramos; Ruby; Vembu Shanthi; Higinio Ramos

doi:10.3934/math.2025313

AIMS Mathematics

2025, Volume 10, Issue 3: 6827-6852. doi: 10.3934/math.2025313

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A numerical approach to approximate the solution of a quasilinear singularly perturbed parabolic convection diffusion problem having a non-smooth source term

1.
Department of Mathematics, National Institute of Technology, Tiruchirappalli, Tamilnadu, India
2.
Scientific Computing Group, University of Salamanca, 49029 Zamora, Spain
3.
Escuela Politécnica Superior de Zamora, Universidad de Salamanca, Avda. Requejo 33, 49029 Zamora, Spain

Received: 07 October 2024 Revised: 18 March 2025 Accepted: 20 March 2025 Published: 26 March 2025
MSC : 35B25, 65N06, 65N12, 65N15

The objective of the present paper is to solve a one-dimensional quasilinear parabolic singularly perturbed problem with a discontinuous source term. Due to the presence of such a discontinuity, an interior layer exists at the location of the discontinuity. The problem is solved by discretizing the spatial variable on a piecewise uniform Shishkin mesh using the standard upwind approach, while the backward Euler scheme is employed on a uniform mesh to discretize the time variable. The method is $ \varepsilon $-uniformly convergent, providing first-order convergence in the time domain and almost first-order convergence in the spatial variable. To validate the theoretical findings, the scheme was tested by numerically solving two examples.
- convection diffusion,
- quasilinear,
- singular perturbation,
- Shishkin mesh,
- standard upwind scheme,
- backward Euler method,
- discontinuous source term,
- weak interior layer
Citation: Ruby, Vembu Shanthi, Higinio Ramos. A numerical approach to approximate the solution of a quasilinear singularly perturbed parabolic convection diffusion problem having a non-smooth source term[J]. AIMS Mathematics, 2025, 10(3): 6827-6852. doi: 10.3934/math.2025313

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Abstract

The objective of the present paper is to solve a one-dimensional quasilinear parabolic singularly perturbed problem with a discontinuous source term. Due to the presence of such a discontinuity, an interior layer exists at the location of the discontinuity. The problem is solved by discretizing the spatial variable on a piecewise uniform Shishkin mesh using the standard upwind approach, while the backward Euler scheme is employed on a uniform mesh to discretize the time variable. The method is $ \varepsilon $-uniformly convergent, providing first-order convergence in the time domain and almost first-order convergence in the spatial variable. To validate the theoretical findings, the scheme was tested by numerically solving two examples.

References

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