Speaker: 张敏助理研究员，北京大学

Inviter: 陈绍示（中国科学院数学机械化重点实验室）

Title: A well-balanced moving mesh discontinuous Galerkin method for shallow water equations and Ripa model on triangular meshes

Language: Chinese

Time & Venue: 2024.06.20 10:00-11:00 思源楼525

Abstract: In this talk a rezoning-type adaptive moving mesh discontinuous Galerkin method is presented for the numerical solution of the shallow water equations and Ripa model with non-flat bottom topography. The Ripa model is a generalization of the shallow water equations that takes temperature variance into the modeling. The well-balance property is crucial to the simulation of perturbation waves over the lake-at-rest steady state such as waves on a lake and tsunami waves in the deep ocean. To ensure the well-balance and positivity-preservation properties, strategies are discussed in the use of slope limiting, positivity-preservation limiting, and data transferring between meshes. Particularly, it is suggested that a DG-interpolation scheme be used for the interpolation of both the flow variables and bottom topography from the old mesh to the new one and after each application of the positivity-preservation limiting on the water depth, a high-order correction be made to the approximation of the bottom topography according to the modifications in the water depth. Mesh adaptivity is realized using an MMPDE moving mesh method with a metric tensor based on an equilibrium variable and water depth. A motivation for this choice of the metric tensor is to adapt the mesh according to both the perturbations of the lake-at-rest steady state and the water depth distribution (bottom topography structure). Numerical examples of both the shallow water equations and Ripa model 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-state. They also show that the mesh adaptation based on the equilibrium variable and water depth give more accurate results than that based on the commonly used entropy function.