A well-balanced positivity-preserving quasi-Lagrange moving mesh DG method for the shallow water equations

TL;DR

This paper introduces a high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method for shallow water equations, effectively handling complex bottom topography and flow features with adaptive mesh movement.

Contribution

It develops a novel moving mesh DG scheme that maintains well-balance and positivity without mesh interpolation, improving accuracy and adaptability over existing rezoning methods.

Findings

Successfully preserves steady states and positivity.

Demonstrates high-order accuracy in 1D and 2D tests.

Effectively adapts mesh to flow features.

Abstract

A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of an update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography.