Abstract
In this paper, we have developed a novel three step second derivative block method and coupled it with fourth order standard compact finite difference schemes for solving time dependent nonlinear partial differential equations (PDEs) of physical relevance. Two well-known problems viz. the FitzHugh–Nagumo equation and the Burgers’ equation have been considered as test problems to check the effectiveness of the proposed scheme. Firstly, we developed a novel block scheme and discussed its characteristics for solving initial-value systems, such as the one resulting from the discretization of the spatial derivatives that appear in the PDEs. Although many time integration techniques already exist to solve discretized PDEs, our goal is to develop a numerical scheme keeping in mind saving computational time while maintaining good accuracy. The proposed block scheme has been proved to be \({\mathcal {A}}\)-stable and consistent. The method performs well for solving the stiff case of the FitzHugh–Nagumo equation, as well as for solving the Burgers equation at different values of viscosity and time. The numerical experiments reveal that the developed numerical scheme is computationally efficient.
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Acknowledgements
We would like to thank the anonymous reviewers for their valuable comments that have greatly contributed to improve the quality of the manuscript. Akansha Mehta would like to thank I. K. Gujral Punjab Technical University Jalandhar, Punjab (India) for providing research facilities for the present work.
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Communicated by Fabricio Simeoni de Sousa.
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Mehta, A., Singh, G. & Ramos, H. Numerical solution of time dependent nonlinear partial differential equations using a novel block method coupled with compact finite difference schemes. Comp. Appl. Math. 42, 201 (2023). https://doi.org/10.1007/s40314-023-02345-3
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DOI: https://doi.org/10.1007/s40314-023-02345-3