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A high-order adaptive numerical algorithm for fractional diffusion wave equation on non-uniform meshes

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Abstract 

In this work, our motivation is to design an impressive new numerical approximation on non-uniform grid points for the Caputo fractional derivative in time \({~}^{\mathrm {C}}_{0}\mathcal {D}_{t}^{\alpha }\) with the order α ∈ (1,2). An adaptive high-order stable implicit difference scheme is developed for the time-fractional diffusion wave equations (TFDWEs) by using estimation of order \(\mathcal {O}(N_{t}^{\alpha -5})\) for the Caputo derivative in the time domain on non-uniform mesh and well-known second-order central difference approximation for estimating the spatial derivative on a uniform mesh. The designed algorithm allows one to build adaptive nature where the scheme is adjusted according to the behaviour of α in order to keep the numerical errors very small and converge to the solution very fast as compared to the previously investigated scheme. We rigorously analyze the local truncation errors, unconditional stability of the proposed method, and its convergence of (5 − α)-th order in time and second-order in space for all values of α ∈ (1,2). A reduced order technique is implemented by using moving mesh refinement and assemble with the derived scheme in order to improve the temporal accuracy at several starting time levels. Furthermore, the numerical stability of the derived adaptive scheme is verified by imposing random external noises. Some numerical tests are given to show that the numerical results are consistent with the theoretical results.

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Acknowledgements

The authors sincerely thank the Editor for taking time to handle the manuscript and two reviewers for carefully reading the manuscript and their constructive comments and suggestions that really improved the quality of the manuscript.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005, India

    Rahul Kumar Maurya & Vineet Kumar Singh

  2. Department of Mathematics, Government Tilak PG College, Katni, 483501, India

    Rahul Kumar Maurya

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Maurya, R.K., Singh, V.K. A high-order adaptive numerical algorithm for fractional diffusion wave equation on non-uniform meshes. Numer Algor 92, 1905–1950 (2023). https://doi.org/10.1007/s11075-022-01372-1

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