The Recovered Space Advection Scheme for Lowest-Order Compatible Finite Element Methods
This work provides a method to obtain higher-order accuracy with lower-order elements, which is beneficial for computational efficiency in geophysical fluid dynamics simulations.
The paper presents a new compatible finite element advection scheme for the compressible Euler equations that achieves second-order accuracy using lowest-order finite element spaces by recovering functions in higher-order spaces. Numerical tests demonstrate the scheme's stability and boundedness.
We present a new compatible finite element advection scheme for the compressible Euler equations. Unlike the discretisations described in Cotter and Kuzmin (2016) and Shipton et al (2018), the discretisation uses the lowest-order family of compatible finite element spaces, but still retains second-order numerical accuracy. This scheme obtains this second-order accuracy by first `recovering' the function in higher-order spaces, before using the discontinuous Galerkin advection schemes of Cotter and Kuzmin (2016). As well as describing the scheme, we also present its stability properties and a strategy for ensuring boundedness. We then demonstrate its properties through some numerical tests, before presenting its use within a model solving the compressible Euler equations.