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Efficient linear implicit invariant-preserving fourth-order numerical scheme for damped nonlinear fourth-order wave equation

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  • Xie, Jianqiang
  • Zhang, Zhiyue

Abstract

In this study, an invariant-preserving linear implicit finite difference method (LIFDM) for damped nonlinear fourth-order wave equation (DNFOWE) is constructed. Firstly, by introducing the auxiliary variables (AVs), the DNFOWE is transformed into an equivalent system of equations with a first-order temporal derivative. Then a decoupled linear implicit invariant-preserving fourth-order difference scheme is designed for the resulting equations. The strategy decreases the amount of storage demanded for the unknowns to be calculated thereby enhancing the efficiency. Subsequently, the discrete invariant property and theoretical analyses of the present algorithm are explored. To our knowledge, the present work marks the first instance of deriving unconditional error estimates for the unknown variables in discrete L∞-norm via energy analysis and mathematical induction. The main novelty of the scheme is invariant-preserving, linear implicit decoupled and suitable for long-duration computations. Eventually, some representative examples are showcased to substantiate the theoretical findings and dynamic behaviors of the presented algorithm.

Suggested Citation

  • Xie, Jianqiang & Zhang, Zhiyue, 2026. "Efficient linear implicit invariant-preserving fourth-order numerical scheme for damped nonlinear fourth-order wave equation," Applied Mathematics and Computation, Elsevier, vol. 516(C).
    • Handle: RePEc:eee:apmaco:v:516:y:2026:i:c:s0096300325006058
    DOI: 10.1016/j.amc.2025.129880
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    References listed on IDEAS

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    1. Achouri, Talha, 2019. "Conservative finite difference scheme for the nonlinear fourth-order wave equation," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 121-131.
    2. Zhang, Gengen, 2021. "Two conservative and linearly-implicit compact difference schemes for the nonlinear fourth-order wave equation," Applied Mathematics and Computation, Elsevier, vol. 401(C).
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    1. Zhang, Gengen, 2021. "Two conservative and linearly-implicit compact difference schemes for the nonlinear fourth-order wave equation," Applied Mathematics and Computation, Elsevier, vol. 401(C).

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