The ‘recovered space’ advection scheme for lowest-order compatible finite element methods

• Published in: Journal of computational physics Vol. 390; pp. 342 - 358
• Main Authors: Bendall, Thomas M., Cotter, Colin J., Shipton, Jemma
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.08.2019; Elsevier Science Ltd
• Subjects: Accuracy; Advection; Advection scheme; Compatibility; Compatible finite element methods; Compressibility; Computational physics; Discontinuous Galerkin; Euler-Lagrange equation; Finite element method; Galerkin method; Mathematical analysis; Numerical weather prediction; Discontinuous Galerkin; Compatible finite element methods; Advection scheme; Numerical weather prediction
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2019.04.013

•A new advection scheme is proposed for lowest-order compatible finite element spaces.•This advection scheme has second order numerical accuracy.•A proof of stability of the advection scheme is provided.•Properties of the advection scheme are investigated. 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.