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Uniformly convergent computational method for singularly perturbed time delayed parabolic differential-difference equations

Jugal Mohapatra (Department of Mathematics, National Institute of Technology Rourkela, Rourkela, India)
Sushree Priyadarshana (Department of Mathematics, National Institute of Technology Rourkela, Rourkela, India)
Narahari Raji Reddy (Kakatiya Institute of Technology and Science, Warangal, India)

Engineering Computations

ISSN: 0264-4401

Article publication date: 26 May 2023

Issue publication date: 2 June 2023

134

Abstract

Purpose

The purpose of this work is to introduce an efficient, global second-order accurate and parameter-uniform numerical approximation for singularly perturbed parabolic differential-difference equations having a large lag in time.

Design/methodology/approach

The small delay and advance terms in spatial direction are handled with Taylor's series approximation. The Crank–Nicholson scheme on a uniform mesh is applied in the temporal direction. The derivative terms in space are treated with a hybrid scheme comprising the midpoint upwind and the central difference scheme at appropriate domains, on two layer-resolving meshes namely, the Shishkin mesh and the Bakhvalov–Shishkin mesh. The computational effectiveness of the scheme is enhanced by the use of the Thomas algorithm which takes less computational time compared to the usual Gauss elimination.

Findings

The proposed scheme is proved to be second-order accurate in time and to be almost second-order (up to a logarithmic factor) uniformly convergent in space, using the Shishkin mesh. Again, by the use of the Bakhvalov–Shishkin mesh, the presence of a logarithmic effect in the spatial-order accuracy is prevented. The detailed analysis of the convergence of the fully discrete scheme is thoroughly discussed.

Research limitations/implications

The use of second-order approximations in both space and time directions makes the complete finite difference scheme a robust approximation for the considered class of model problems.

Originality/value

To validate the theoretical findings, numerical simulations on two different examples are provided. The advantage of using the proposed scheme over some existing schemes in the literature is proved by the comparison of the corresponding maximum absolute errors and rates of convergence.

Keywords

Acknowledgements

The second author S. Priyadarshana expresses her sincere thanks to the Department of Science & Technology, Government of India, for providing INSPIRE fellowship (IF 180938).

Citation

Mohapatra, J., Priyadarshana, S. and Raji Reddy, N. (2023), "Uniformly convergent computational method for singularly perturbed time delayed parabolic differential-difference equations", Engineering Computations, Vol. 40 No. 3, pp. 694-717. https://doi.org/10.1108/EC-06-2022-0396

Publisher

:

Emerald Publishing Limited

Copyright © 2023, Emerald Publishing Limited

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