In this paper, we propose a class of adaptive multiresolution (also called adaptive sparse grid) ultra-weak discontinuous Galerkin (UWDG) methods for solving some nonlinear dispersive wave equations including the Korteweg-de Vries (KdV) equation and its two dimensional generalization, the Zakharov-Kuznetsov (ZK) equation. The UWDG formulation, which relies on repeated integration by parts, was proposed for KdV equation in \cite{cheng2008discontinuous}. For the ZK equation which contains mixed derivative terms, we develop a new UWDG formulation. The $L^2$ stability and the optimal error estimate with a novel local projection are established for this new scheme on regular meshes. Adaptivity is achieved based on multiresolution and is particularly effective for capturing solitary wave structures. Various numerical examples are presented to demonstrate the accuracy and capability of our methods.

翻译：在本文中,我们提出了一类适应性多分辨率(又称适应性稀疏网格)超弱不连续加勒金(UWDG)方法,用于解决某些非线性分散波方程式,包括Korteweg-de Vries(KdV)方程式及其两个维维通用方程式Zakharov-Kuznetsov(ZK)方程式。UWDG配方法依靠各部分的反复整合,在\cite{cheng2008dunity}中为KdV方程式提出。对于包含混合衍生物术语的ZK方程式,我们开发了一种新的UWDG方程式。对于这个新办法的精度和能力,我们用新的本地预测确定了$2的稳定性和最佳误差估计值。适应性是在多分辨率的基础上实现的,对于捕捉孤立的波结构特别有效。提供了各种数字例子,以证明我们方法的准确性和能力。