High‐order compact finite difference method for solving the two‐dimensional fourth‐order nonlinear hyperbolic equation is considered in this article. In order to design an implicit compact finite difference scheme, the fourth‐order equation is written as a system of two second‐order equations by introducing the second‐order spatial derivative as a new variable. The second‐order spatial derivatives are approximated by the compact finite difference operators to obtain a fourth‐order convergence. As well as, the second‐order time derivative is approximated by the central difference method. Then, existence and uniqueness of numerical solution is given. The stability and convergence of the compact finite difference scheme are proved by the energy method. Numerical results are provided to verify the accuracy and efficiency of this scheme.

中文翻译：

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二维四阶非线性双曲方程的紧致差分格式

本文考虑了求解二维四阶非线性双曲方程的高阶紧致有限差分法。为了设计隐式紧致有限差分方案，通过引入二阶空间导数作为新变量，将四阶方程写成两个二阶方程的系统。紧致的有限差分算子对二阶空间导数进行近似，以获得四阶收敛。同样，二阶时间导数也可以通过中心差分法进行近似。然后给出了数值解的存在性和唯一性。通过能量法证明了紧致有限差分格式的稳定性和收敛性。提供数值结果以验证该方案的准确性和效率。