Second Order Finite Volume Scheme for Shallow Water Equations on Manifolds

TL;DR

This paper introduces a second-order finite volume scheme for solving shallow water equations on manifolds, leveraging covariant coordinates and metric tensors to handle complex geometries and compute curvature dynamically.

Contribution

It presents a novel second-order accurate scheme that incorporates metric tensors in covariant coordinates, enabling automatic curvature computation and flexible modeling on manifolds.

Findings

The scheme achieves second-order accuracy in numerical tests.

It effectively models shallow water flows on complex geometries.

The method allows dynamic curvature computation from physical variables.

Abstract

In this work, we propose a second-order accurate scheme for shallow water equations in general covariant coordinates over manifolds. In particular, the covariant parametrization in general covariant coordinates is induced by the metric tensor associated to the manifold. The model is then re-written in a hyperbolic form with a tuple of conserved variables composed both of the evolving physical quantities and the metric coefficients. This formulation allows the numerical scheme to i) automatically compute the curvature of the manifold as long as the physical variables are evolved and ii) numerically study complex physical domains over simple computational domains.