In this paper, the local discontinuous Galerkin method is developed to solve the two-dimensional Camassa–Holm equation in rectangular meshes. The idea of LDG methods is to suitably rewrite a higher-order partial differential equations into a first-order system, then apply the discontinuous Galerkin method to the system. A key ingredient for the success of such methods is the correct design of interface numerical fluxes. The energy stability for general solutions of the method is proved. Comparing with the Camassa–Holm equation in one-dimensional case, there are more auxiliary variables which are introduced to handle high-order derivative terms. The proof of the stability is more complicated. The resulting scheme is high-order accuracy and flexible for arbitrary h and p adaptivity. Different types of numerical simulations are provided to illustrate the accuracy and stability of the method.

中文翻译：

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二维Camassa-Holm方程的局部不连续Galerkin方法

在本文中，开发了局部不连续Galerkin方法来求解矩形网格中的二维Camassa-Holm方程。LDG方法的思想是将一个高阶偏微分方程适当地重写为一阶系统，然后将不连续Galerkin方法应用于该系统。此类方法成功的关键因素是界面数值通量的正确设计。证明了该方法一般解的能量稳定性。与一维情况下的Camassa–Holm方程相比，引入了更多辅助变量来处理高阶导数项。稳定性的证明更为复杂。生成的方案具有高阶精度，并且对于任意h和p都具有灵活性适应性。提供了不同类型的数值模拟，以说明该方法的准确性和稳定性。