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An Effective Dissipation-Preserving Fourth-Order Difference Solver for Fractional-in-Space Nonlinear Wave Equations

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Abstract

In this paper, we devise an efficient dissipation-preserving fourth-order difference solver for the fractional-in-space nonlinear wave equations. First of all, we present a detailed derivation of the discrete energy dissipation property of the system. Then, with the help of the mathematical induction and Brouwer fixed point theorem, it is shown that the proposed scheme is uniquely solvable. Subsequently, by virtue of utilizing the discrete energy method, it is proven that the proposed solver achieves the convergence rates of \({\mathcal {O}}(\Delta t^2+h^{4})\) in the discrete \(L^{\infty }\)- norm, and is unconditionally stable. And moreover, the exhibited convergence analysis is unconditional for the time step and space size, in comparison with the restrictive conditions required in the existing works. Finally, numerical results confirm the efficiency of the proposed scheme and exhibit the correctness of theoretical results.

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Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11471166, 11401294), Natural Science Foundation of Jiangxi Provincial Education Department (Grant No. GJJ160706), State Scholarship Fund of CSC for Overseas Studies (Grant No. 201806860014). The authors greatly appreciate the anonymous referees for their valuable comments and suggestions, which have improved the quality of this paper.

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  1. School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing, 210023, China

    Jianqiang Xie & Zhiyue Zhang

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  1. Jianqiang Xie
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  2. Zhiyue Zhang
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Correspondence to Zhiyue Zhang.

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Xie, J., Zhang, Z. An Effective Dissipation-Preserving Fourth-Order Difference Solver for Fractional-in-Space Nonlinear Wave Equations. J Sci Comput 79, 1753–1776 (2019). https://doi.org/10.1007/s10915-019-00921-6

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  • DOI: https://doi.org/10.1007/s10915-019-00921-6

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