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A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations

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  • Macías-Díaz, J.E.
  • Hendy, A.S.
  • De Staelen, R.H.

Abstract

In this work, we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the Riesz type. The model under consideration generalizes various nonlinear wave equations, including the sine-Gordon and the nonlinear Klein–Gordon models. The system considered in this work is conservative when homogeneous Dirichlet boundary conditions are imposed. Motivated by this fact, we propose a finite-difference method based on fractional centered differences that is capable of preserving the discrete energy of the system. The method under consideration is a nonlinear implicit scheme which has various numerical properties. Among the most interesting numerical features, we show that the methodology is consistent of second order in time and fourth order in space. Moreover, we show that the technique is stable and convergent. Some numerical simulations show that the method is capable of preserving the energy of the discrete system. This characteristic of the technique is in obvious agreement with the properties of its continuous counterpart.

Suggested Citation

  • Macías-Díaz, J.E. & Hendy, A.S. & De Staelen, R.H., 2018. "A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 1-14.
    • Handle: RePEc:eee:apmaco:v:325:y:2018:i:c:p:1-14
    DOI: 10.1016/j.amc.2017.12.002
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    References listed on IDEAS

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    1. Qin, Wendi & Ding, Deqiong & Ding, Xiaohua, 2015. "Two boundedness and monotonicity preserving methods for a generalized Fisher-KPP equation," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 552-567.
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    Cited by:

    1. Wu, Longbin & Ma, Qiang & Ding, Xiaohua, 2021. "Energy-preserving scheme for the nonlinear fractional Klein–Gordon Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 1110-1129.
    2. Shaojun Chen & Yayun Fu, 2024. "Linearly Implicit Conservative Schemes with a High Order for Solving a Class of Nonlocal Wave Equations," Mathematics, MDPI, vol. 12(15), pages 1-13, August.
    3. Xing, Zhiyong & Wen, Liping & Wang, Wansheng, 2021. "An explicit fourth-order energy-preserving difference scheme for the Riesz space-fractional Sine–Gordon equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 624-641.
    4. Zhao, Jingjun & Li, Yu & Xu, Yang, 2019. "An explicit fourth-order energy-preserving scheme for Riesz space fractional nonlinear wave equations," Applied Mathematics and Computation, Elsevier, vol. 351(C), pages 124-138.
    5. Macías-Díaz, J.E., 2018. "A numerically efficient Hamiltonian method for fractional wave equations," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 231-248.
    6. Wang, Nan & Shi, Dongyang, 2021. "Two efficient spectral methods for the nonlinear fractional wave equation in unbounded domain," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 696-718.

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