In this paper, an ultra-weak local discontinuous Galerkin (UWLDG) method for a class of nonlinear fourth-order wave equations is designed and analyzed. The UWLDG method is a new DG method designed for solving partial differential equations (PDEs) with high order spatial derivatives. We prove the energy conserving property of our scheme and its optimal error estimates in the $L^2$-norm for the solution itself as well as for the auxiliary variables approximating the derivatives of the solution. Compatible high order energy conserving time integrators are also proposed. The theoretical results are confirmed by numerical experiments.

中文翻译：

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非线性四阶波动方程的不连续Galerkin方法及其误差估计

针对一类非线性四阶波动方程，设计并分析了一种超弱局部不连续伽勒金方法。UWLDG方法是一种新的DG方法，设计用于求解具有高阶空间导数的偏微分方程（PDE）。我们证明了该方案的节能特性及其最优误差估计。${\mathrm{大号}}^2$-范数用于解决方案本身以及用于逼近解决方案导数的辅助变量。还提出了兼容的高阶节能时间积分器。理论结果通过数值实验得到证实。