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Convergence and Stability in Maximum Norms of Linearized Fourth-Order Conservative Compact Scheme for Benjamin–Bona–Mahony–Burgers’ Equation

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Abstract

In the paper, a newly developed three-point fourth-order compact operator is utilized to construct an efficient compact finite difference scheme for the Benjamin–Bona–Mahony–Burgers’ (BBMB) equation. The detailed derivation is carried out based on the reduction order method together with a three-level linearized technique. The conservative invariant, boundedness and unique solvability are studied at length. The convergence is proved by the technical energy argument and induction method with the optimal convergence order \(\mathcal {O}(\tau ^2+h^4)\) in the sense of the maximum norm. The stability under mild conditions can be achieved based on the uniform boundedness of the numerical solution. The present scheme is very efficient in practical computation since only a linear system needs to be solved at each time. The extensive numerical examples verify our theoretical results and demonstrate the scheme’s superiority when compared with state-of-the-art those in the references.

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Data Availability Statement

All data or codes generated or used during the study are available from the corresponding author by request.

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Acknowledgements

The first author thanks Professor Zhi-zhong Sun for his helpful discussion. The authors are very grateful to the anonymous referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of the paper.

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  1. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Qifeng Zhang & Lingling Liu

  2. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Qifeng Zhang

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  1. Qifeng Zhang
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Correspondence to Qifeng Zhang.

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Qifeng Zhang was supported in part by Natural Science Foundation of China (No. 11501514), in part by the Natural Sciences Foundation of Zhejiang Province under Grant LY19A010026, in part by project funded by China Postdoctoral Science Foundation under Grant 2018M642131 when he studied in Southeast University.

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Zhang, Q., Liu, L. Convergence and Stability in Maximum Norms of Linearized Fourth-Order Conservative Compact Scheme for Benjamin–Bona–Mahony–Burgers’ Equation. J Sci Comput 87, 59 (2021). https://doi.org/10.1007/s10915-021-01474-3

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