Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements

TL;DR

This paper introduces two novel schemes to enhance the spatial accuracy of discretising the vector transport equation using lowest-order quadrilateral Raviart-Thomas finite elements on curved manifolds, validated through numerical tests.

Contribution

It presents higher-order reconstruction and mixed finite element schemes, including a SUPG stabilization, to improve accuracy of vector transport discretisation on curved geometries.

Findings

Enhanced accuracy demonstrated in numerical tests.

Effective stabilization via SUPG method.

Improved transport of vector fields on curved manifolds.

Abstract

Within finite element models of fluids, vector-valued fields such as velocity or momentum variables are commonly discretised using the Raviart-Thomas elements. However, when using the lowest-order quadrilateral Raviart-Thomas elements, standard finite element discretisations of the vector transport equation typically have a low order of spatial accuracy. This paper describes two schemes that improve the accuracy of transporting such vector-valued fields on two-dimensional curved manifolds. The first scheme that is presented reconstructs the transported field in a higher-order function space, where the transport equation is then solved. The second scheme applies a mixed finite element formulation to the vector transport equation, simultaneously solving for the transported field and its vorticity. An approach to stabilising this mixed vector-vorticity formulation is presented that uses a Streamline Upwind Petrov-Galerkin (SUPG) method. These schemes are then demonstrated, along with their accuracy properties, through some numerical tests. Two new test cases are used to assess the transport of vector-valued fields on curved manifolds, solving the vector transport equation in isolation. The improvement of the schemes is also shown through two standard test cases for rotating shallow-water models.