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Fully-discrete energy-preserving scheme for the space-fractional Klein–Gordon equation via Lagrange multiplier type scalar auxiliary variable approach

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  • Huang, Qiong-Ao
  • Zhang, Gengen
  • Wu, Bing

Abstract

A family of effective fully-discrete energy-preserving schemes for the space-fractional Klein–Gordon equation is developed in this paper. First, the recently developed Lagrange multiplier type scalar auxiliary variable approach is employed to obtain a new equivalent system from the original space-fractional Klein–Gordon system. Then, a family of special second-order implicit, explicit and implicit approximations to respectively discretize the linear parts, nonlinear parts and time-derivative parts are obtained in the above equivalent system to establish a family of semi-discrete (continuous in space) energy-preserving schemes. Furthermore, the Fourier pseudo-spectral method is used to discretize the space for extending to the fully-discrete case and rigorous theoretical proofs guarantee its conservation of original energy. Especially, the well-known implicit–explicit Crank–Nicolson type scheme is only one of the above-mentioned schemes. It is inspiring that the main computational efforts of this method in each time step are only to solve two linear, decoupled differential equations with constant coefficients different from non-homogeneous terms, which thus can be effectively solved. Finally, numerical experiments are carried out to verify the theoretical results of the accuracy, efficiency and conservation of original energy.

Suggested Citation

  • Huang, Qiong-Ao & Zhang, Gengen & Wu, Bing, 2022. "Fully-discrete energy-preserving scheme for the space-fractional Klein–Gordon equation via Lagrange multiplier type scalar auxiliary variable approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 265-277.
    • Handle: RePEc:eee:matcom:v:192:y:2022:i:c:p:265-277
    DOI: 10.1016/j.matcom.2021.09.002
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    References listed on IDEAS

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    1. Nagy, A.M., 2017. "Numerical solution of time fractional nonlinear Klein–Gordon equation using Sinc–Chebyshev collocation method," Applied Mathematics and Computation, Elsevier, vol. 310(C), pages 139-148.
    2. Hosseini, Kamyar & Ilie, Mousa & Mirzazadeh, Mohammad & Yusuf, Abdullahi & Sulaiman, Tukur Abdulkadir & Baleanu, Dumitru & Salahshour, Soheil, 2021. "An effective computational method to deal with a time-fractional nonlinear water wave equation in the Caputo sense," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 248-260.
    3. Hu, Dongdong & Cai, Wenjun & Xu, Zhuangzhi & Bo, Yonghui & Wang, Yushun, 2021. "Dissipation-preserving Fourier pseudo-spectral method for the space fractional nonlinear sine–Gordon equation with damping," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 35-59.
    4. Wang, Junjie & Xiao, Aiguo, 2019. "Conservative Fourier spectral method and numerical investigation of space fractional Klein–Gordon–Schrödinger equations," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 348-365.
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