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Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel

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Abstract

A spatial two-grid compact difference method for the nonlinear Volterra integro-differential equations with the Abel kernel is proposed to reduce the computational cost and improve the accuracy of the scheme. The proposed scheme firstly solves a small nonlinear compact finite difference system on a coarse grid and then solves a large linear compact finite difference system on a fine grid based on the coarse-grid solution using Newton’s linearization and the higher-order mapping operator. Then, we combine the properties of the higher-order mapping operator with the two-grid analysis method as well as the singularity of the solutions to prove the stability and convergence of the proposed algorithm under the \(L^2\)-norm with the order \(O(\tau ^2+H^8+h^4)\). Moreover, the constructed method is extended to the two-dimensional case. Several numerical experiments verify the effectiveness of the proposed method and show its competitiveness compared with the existing methods.

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Acknowledgements

The authors are grateful for helpful suggestions from the reviewers.

Funding

This work was partially supported by the Taishan Scholars Program of Shandong Province (No. tsqn202306083), the National Natural Science Foundation of China (No. 12301555), and the National Key R &D Program of China (No. 2023YFA1008903).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, 410081, People’s Republic of China

    Hao Chen & Wenlin Qiu

  2. Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia

    Mahmoud A. Zaky

  3. Department of Applied Mathematics, National Research Centre, Dokki, Cairo, 12622, Egypt

    Mahmoud A. Zaky

  4. School of Mathematics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China

    Xiangcheng Zheng & Wenlin Qiu

  5. Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg, 620002, Russia

    Ahmed S. Hendy

  6. Department of Mathematics, Faculty of Science, Benha University, Benha, 13511, Egypt

    Ahmed S. Hendy

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  1. Hao Chen
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  2. Mahmoud A. Zaky
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  3. Xiangcheng Zheng
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  4. Ahmed S. Hendy
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  5. Wenlin Qiu
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Contributions

X. Zheng: Funding acquisition, writing—review and editing, writing—original draft.

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Correspondence to Wenlin Qiu.

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Chen, H., Zaky, M.A., Zheng, X. et al. Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel. Numer Algor 98, 677–718 (2025). https://doi.org/10.1007/s11075-024-01811-1

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