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Nonlinear & linearized compact difference schemes and Richardson extrapolation for nonlinear delay Sobolev equations with second-order derivative

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  • Wang, Kun
  • Wang, Bo

Abstract

Delay Sobolev equations (DSEs) play a crucial role in disciplines such as fluid dynamics and thermodynamics. This paper primarily focuses on the theoretical analysis and numerical computation of the initial–boundary value problem for the nonlinear DSEs with second-order derivative (DSEs-SOD). We propose four distinct schemes in total, including the nonlinear compact difference scheme (NCDS), linearized compact difference scheme (LCDS), Richardson extrapolation based on the nonlinear Crank–Nicolson scheme (ENCNS) and Richardson extrapolation based on the linearized Crank–Nicolson scheme (ELCNS). Additionally, we rigorously analyze the convergence of these schemes under appropriate conditions, the former two compact schemes achieve fourth-order in space and second-order in time. Moreover, the latter two extrapolation schemes are more effective, they satisfy fourth-order convergence in both time and space. Numerical experiment is performed to verify the theoretical convergence rates and to evaluate the computational efficiency of the proposed numerical schemes.

Suggested Citation

  • Wang, Kun & Wang, Bo, 2026. "Nonlinear & linearized compact difference schemes and Richardson extrapolation for nonlinear delay Sobolev equations with second-order derivative," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 245(C), pages 713-731.
    • Handle: RePEc:eee:matcom:v:245:y:2026:i:c:p:713-731
    DOI: 10.1016/j.matcom.2026.03.004
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    References listed on IDEAS

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    1. Tang, Changyang & Zhang, Chengjian, 2021. "A fully discrete θ-method for solving semi-linear reaction–diffusion equations with time-variable delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 179(C), pages 48-56.
    2. Li, Xiaoli & Rui, Hongxing, 2019. "A block-centered finite difference method for the nonlinear Sobolev equation on nonuniform rectangular grids," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
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