A Class of Adaptive Multiresolution Ultra-Weak Discontinuous Galerkin Methods for Some Nonlinear Dispersive Wave Equations

Abstract

In this paper, we propose a class of adaptive multiresolution (also called the 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 the KdV equation in [7]. For the ZK equation, which contains mixed derivative terms, we develop a new UWDG formulation. The $L^2$ stability is established for this new scheme on regular meshes, and the optimal error estimate with a novel local projection is obtained for a simplified ZK equation. 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.