2021/04/12 by Juntao Huang, Yong Liu, Huang, Juntao +7
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #Numerical methods for differential equations #cs.NA #math.NA #physics.comp-ph
paper · pdf · doi:10.48550/arxiv.2104.05523
arxiv created 2021/04/12 · arxiv updated 2021/04/13
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 \citecheng2008discontinuous. For the ZK equation which contains mixed derivative terms, we develop a new UWDG formulation. The L2 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.