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A time two-grid difference method for nonlinear generalized viscous Burgers’ equation

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Abstract

In this paper, we investigate a time two-grid algorithm to get the numerical solution of nonlinear generalized viscous Burgers’ equation, which is the first time that the two-grid method is used to solve this problem. Based on Crank–Nicolson finite difference scheme, we establish the time two-grid difference (TTGD) scheme which consists of three computational procedures to reduce the computational cost compared with the general finite difference (GFD) scheme. The cut-offf function method is applied to prove the conservation, unique solvability, the prior estimate and convergence in \(L^2\)-norm and \(L^{\infty }\)-norm of the TTGD scheme on the coarse grid and fine grid, respectively. Comparing our TTGD scheme with GFD scheme in Zhang et al. (Appl Math Lett 112:106719, 2021), we provided the proof the uniqueness the solution of the nonlinear scheme, direct proof of convergence in \(L^2\)-norm and the prior estimate both on the coarse mesh and fine mesh. The numerical results show that our TTGD scheme is more efficient than the GDF scheme in Zhang et al. (2021) in terms of the CPU time. Particularly, our method not only improves the efficiency, but also preserves the energy conservation of the original model.

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Acknowledgements

The authors are grateful to the referees for many helpful suggestions.

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  1. School of Science, Hunan University of Technology, Zhuzhou, 412007, China

    Yang Shi & Xuehua Yang

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  1. Yang Shi
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  2. Xuehua Yang
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Correspondence to Xuehua Yang.

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The work was supported by National Natural Science Foundation of China Mathematics Tianyuan Foundation (12226337, 12226340, 12126321), Scientific Research Fund of Hunan Provincial Education Department (21B0550), Hunan Provincial Natural Science Foundation of China (2022JJ50083).

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Shi, Y., Yang, X. A time two-grid difference method for nonlinear generalized viscous Burgers’ equation. J Math Chem (2024). https://doi.org/10.1007/s10910-024-01592-x

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