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

In this paper, we propose a class of adaptive multiresolution (also called the sparse grid) ultra-weak discontinuous Galerkin (UWDG) methods for solving some nonlinear dispersive wave equations including Korteweg--de Vries (KdV) equation and its two-dimensional generalization, Zakharov--Kuznetsov (ZK) equation. The UWDG formulation, which relies on repeated integration by parts, was proposed KdV in [7]. For ZK equation, contains mixed derivative terms, develop new formulation. $L^2$ stability is established scheme regular meshes, optimal error estimate with novel local projection obtained simplified Adaptivity achieved based particularly effective capturing solitary structures. Various numerical examples are presented to demonstrate accuracy capability our methods.