A well-balanced moving mesh discontinuous Galerkin method for the Ripa model on triangular meshes

作     者：Huang, Weizhang Li, Ruo Qiu, Jianxian Zhang, Min 

作者机构：Department of Mathematics University of Kansas LawrenceKS66045 United States CAPT LMAM School of Mathematical Sciences Peking University Beijing100871 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China School of Mathematical Sciences Peking University Beijing100871 China 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2022年

核心收录：

主　　题：Interpolation 

摘      要：A well-balanced moving mesh discontinuous Galerkin (DG) method is proposed for the numerical solution of the Ripa model - a generalization of the shallow water equations that accounts for effects of water temperature variations. Thermodynamic processes are important particularly in the upper layers of the ocean where the variations of sea surface temperature play a fundamental role in climate change. The well-balance property which requires numerical schemes to preserve the lake-at-rest steady state is crucial to the simulation of perturbation waves over that steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance, positivity-preserving, and high-order properties, a DG-interpolation scheme (with or without scaling positivity-preserving limiter) and special treatments pertaining to the Ripa model are employed in the transfer of both the flow variables and bottom topography from the old mesh to the new one and in the TVB limiting process. Mesh adaptivity is realized using an MMPDE moving mesh approach and a metric tensor based on an equilibrium variable and water depth. A motivation is to adapt the mesh according to both the perturbations of the lake-at-rest steady state and the water depth distribution (bottom structure). Numerical examples in one and two dimensions are presented to demonstrate the well-balance, high-order accuracy, and positivity-preserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady *** Codes 65M50, 65M60, 76B15, 35Q35 Copyright © 2022, The Authors. All rights reserved.