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

TL;DR

This paper introduces a high-order, well-balanced moving mesh discontinuous Galerkin method tailored for the Ripa model, effectively capturing small perturbations in water temperature and depth on triangular meshes.

Contribution

It develops a novel DG-based moving mesh scheme that preserves steady states and positivity for the Ripa model, incorporating specialized transfer and limiting techniques.

Findings

Successfully preserves lake-at-rest steady state.

Accurately captures small perturbations and waves.

Demonstrates high-order accuracy and stability in numerical tests.

Abstract

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 state.