Uniformly convergent computational method for singularly perturbed time delayed parabolic differential-difference equations
ISSN: 0264-4401
Article publication date: 26 May 2023
Issue publication date: 2 June 2023
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
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