This paper presents a fully discrete scheme by discretizing the space with the Legendre-Galerkin method and the time with the Crank-Nicolson method to solve the two-dimensional second-order wave equation. Unconditional stability and optimal error estimates in both L² and H¹ norms of the fully discrete Crank-Nicolson Galerkin method are obtained. Numerical results confirm exponential convergence of the proposed method in space and second-order convergence in time. Also, the numerical experiments show the discrete energy conservation and efficiency of long-time numerical calculation.

中文翻译：

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二维二阶波动方程的 Crank-Nicolson Legendre-Galerkin 方法的无条件稳定性和最优误差估计

本文提出了一个完全离散的方案，用Legendre-Galerkin方法离散空间，用Crank-Nicolson方法离散时间来求解二维二阶波动方程。获得了完全离散的 Crank-Nicolson Galerkin 方法的L ²和H ¹范数中的无条件稳定性和最优误差估计。数值结果证实了所提出的方法在空间上的指数收敛和时间上的二阶收敛。此外，数值实验表明了长时间数值计算的离散能量守恒和效率。