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Two-grid finite difference method for 1D fourth-order Sobolev-type equation with Burgers’ type nonlinearity

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  • Chen, Hao
  • Nikan, Omid
  • Qiu, Wenlin
  • Avazzadeh, Zakieh

Abstract

This paper proposes a time two-grid finite difference (TTGFD) technique for computing numerical solution of the one-dimensional (1D) fourth-order Sobolev-type equation with Burgers-type nonlinearity. The proposed strategy mainly contains three computational stages. First, the fully nonlinear problem is approximated over a coarse grid with grid size τC. Second, an approximate solution is obtained on a fine grid with grid size τF based on the solution of the coarse grid using the Lagrangian interpolation formula. Finally, the linear problem is solved on the fine grid. Compared with the standard finite difference (SFD) scheme, an advantage of the TTGFD method is that it can maintain optimal accuracy while reducing the computational cost. Meanwhile, based on the reduced order and the discrete energy method, the conservative invariant and uniqueness of proposed method are demonstrated. In addition, the stability and convergence with the order O(τC2+τF2+h2) are evaluated in the L∞-norm, where h is the spatial step size. Finally, numerical results show the validity of the proposed strategy and support the theoretical results.

Suggested Citation

  • Chen, Hao & Nikan, Omid & Qiu, Wenlin & Avazzadeh, Zakieh, 2023. "Two-grid finite difference method for 1D fourth-order Sobolev-type equation with Burgers’ type nonlinearity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 209(C), pages 248-266.
    • Handle: RePEc:eee:matcom:v:209:y:2023:i:c:p:248-266
    DOI: 10.1016/j.matcom.2023.02.014
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    References listed on IDEAS

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    1. Niu, Yuxuan & Liu, Yang & Li, Hong & Liu, Fawang, 2023. "Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 387-407.
    2. Ramadan, Mohamed A. & El-Danaf, Talaat S. & Abd Alaal, Faisal E.I., 2005. "A numerical solution of the Burgers’ equation using septic B-splines," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 795-804.
    3. Qiu, Wenlin & Chen, Hongbin & Zheng, Xuan, 2019. "An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 298-314.
    4. Ramadan, Mohamed A. & El-Danaf, Talaat S. & Abd Alaal, Faisal E.I., 2005. "A numerical solution of the Burgers’ equation using septic B-splines," Chaos, Solitons & Fractals, Elsevier, vol. 26(4), pages 1249-1258.
    5. Pany, Ambit K. & Bajpai, Saumya & Mishra, Soumyarani, 2020. "Finite element Galerkin method for 2D Sobolev equations with Burgers’ type nonlinearity," Applied Mathematics and Computation, Elsevier, vol. 387(C).
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