Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements
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. In this work, 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.
- Authors:
-
- United Kingdom Meteorological Office, Exeter, Devon (United Kingdom)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Publication Date:
- Research Org.:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)
- OSTI Identifier:
- 1903550
- Report Number(s):
- LA-UR-22-25984
Journal ID: ISSN 0021-9991; TRN: US2311863
- Grant/Contract Number:
- 89233218CNA000001
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Computational Physics
- Additional Journal Information:
- Journal Volume: 474; Journal ID: ISSN 0021-9991
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; Mathematics; Vector transport, Raviart-Thomas finite elements; Recovered finite element method; Vorticity; SUPG
Citation Formats
Bendall, Thomas M., and Wimmer, Golo Albert. Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements. United States: N. p., 2022.
Web. doi:10.1016/j.jcp.2022.111834.
Bendall, Thomas M., & Wimmer, Golo Albert. Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements. United States. https://doi.org/10.1016/j.jcp.2022.111834
Bendall, Thomas M., and Wimmer, Golo Albert. Tue .
"Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements". United States. https://doi.org/10.1016/j.jcp.2022.111834. https://www.osti.gov/servlets/purl/1903550.
@article{osti_1903550,
title = {Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements},
author = {Bendall, Thomas M. and Wimmer, Golo Albert},
abstractNote = {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. In this work, 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.},
doi = {10.1016/j.jcp.2022.111834},
journal = {Journal of Computational Physics},
number = ,
volume = 474,
place = {United States},
year = {Tue Dec 06 00:00:00 EST 2022},
month = {Tue Dec 06 00:00:00 EST 2022}
}
Works referenced in this record:
Recovered finite element methods
journal, April 2018
- Georgoulis, Emmanuil H.; Pryer, Tristan
- Computer Methods in Applied Mechanics and Engineering, Vol. 332
Slate: extending Firedrake's domain-specific abstraction to hybridized solvers for geoscience and beyond
journal, February 2020
- Gibson, Thomas H.; Mitchell, Lawrence; Ham, David A.
- Geoscientific Model Development, Vol. 13, Issue 2
Dispersion analysis of compatible Galerkin schemes on quadrilaterals for shallow water models
journal, June 2019
- Eldred, Christopher; Le Roux, Daniel Y.
- Journal of Computational Physics, Vol. 387
Finite element exterior calculus: from Hodge theory to numerical stability
journal, January 2010
- Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar
- Bulletin of the American Mathematical Society, Vol. 47, Issue 2
High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations
journal, September 2009
- Bernard, P. -E.; Remacle, J. -F.; Comblen, R.
- Journal of Computational Physics, Vol. 228, Issue 17
A mixed finite‐element, finite‐volume, semi‐implicit discretization for atmospheric dynamics: Cartesian geometry
journal, March 2019
- Melvin, Thomas; Benacchio, Tommaso; Shipway, Ben
- Quarterly Journal of the Royal Meteorological Society, Vol. 145, Issue 724
Embedded discontinuous Galerkin transport schemes with localised limiters
journal, April 2016
- Cotter, C. J.; Kuzmin, D.
- Journal of Computational Physics, Vol. 311
The ICON (ICOsahedral Non-hydrostatic) modelling framework of DWD and MPI-M: Description of the non-hydrostatic dynamical core
journal, June 2014
- Zängl, Günther; Reinert, Daniel; Rípodas, Pilar
- Quarterly Journal of the Royal Meteorological Society, Vol. 141, Issue 687
Horizontal grids for global weather and climate prediction models: a review
journal, November 2011
- Staniforth, Andrew; Thuburn, John
- Quarterly Journal of the Royal Meteorological Society, Vol. 138, Issue 662
A standard test case suite for two-dimensional linear transport on the sphere
journal, January 2012
- Lauritzen, P. H.; Skamarock, W. C.; Prather, M. J.
- Geoscientific Model Development, Vol. 5, Issue 3
The ‘recovered space’ advection scheme for lowest-order compatible finite element methods
journal, August 2019
- Bendall, Thomas M.; Cotter, Colin J.; Shipton, Jemma
- Journal of Computational Physics, Vol. 390
A variational $\boldsymbol{H}({\rm div})$ finite-element discretization approach for perfect incompressible fluids
journal, June 2017
- Natale, Andrea; Cotter, Colin J.
- IMA Journal of Numerical Analysis, Vol. 38, Issue 3
Petrov–Galerkin flux upwinding for mixed mimetic spectral elements, and its application to geophysical flow problems
journal, May 2021
- Lee, David
- Computers & Mathematics with Applications, Vol. 89
A compatible finite‐element discretisation for the moist compressible Euler equations
journal, July 2020
- Bendall, Thomas M.; Gibson, Thomas H.; Shipton, Jemma
- Quarterly Journal of the Royal Meteorological Society, Vol. 146, Issue 732
The Met Office Unified Model Global Atmosphere 6.0/6.1 and JULES Global Land 6.0/6.1 configurations
journal, January 2017
- Walters, David; Boutle, Ian; Brooks, Malcolm
- Geoscientific Model Development, Vol. 10, Issue 4
Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements: Energy- and enstrophy-conserving schemes
journal, February 2014
- McRae, Andrew T. T.; Cotter, Colin J.
- Quarterly Journal of the Royal Meteorological Society, Vol. 140, Issue 684
A finite element exterior calculus framework for the rotating shallow-water equations
journal, January 2014
- Cotter, C. J.; Thuburn, J.
- Journal of Computational Physics, Vol. 257
A class of deformational flow test cases for linear transport problems on the sphere
journal, November 2010
- Nair, Ramachandran D.; Lauritzen, Peter H.
- Journal of Computational Physics, Vol. 229, Issue 23
A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids
journal, May 2010
- Ringler, T. D.; Thuburn, J.; Klemp, J. B.
- Journal of Computational Physics, Vol. 229, Issue 9
Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
journal, September 2001
- Cockburn, Bernardo; Shu, Chi-Wang
- Journal of Scientific Computing, Vol. 16, Issue 3, p. 173-261
Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere
journal, December 2018
- Shipton, J.; Gibson, T. H.; Cotter, C. J.
- Journal of Computational Physics, Vol. 375
Edge stabilization for Galerkin approximations of convection–diffusion–reaction problems
journal, April 2004
- Burman, Erik; Hansbo, Peter
- Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue 15-16
Automated Generation and Symbolic Manipulation of Tensor Product Finite Elements
journal, January 2016
- McRae, A. T. T.; Bercea, G. -T.; Mitchell, L.
- SIAM Journal on Scientific Computing, Vol. 38, Issue 5
An initial-value problem for testing numerical models of the global shallow-water equations
journal, January 2004
- Galewsky, Joseph; Scott, Richard K.; Polvani, Lorenzo M.
- Tellus A: Dynamic Meteorology and Oceanography, Vol. 56, Issue 5
Finite element differential forms on curvilinear cubic meshes and their approximation properties
journal, April 2014
- Arnold, Douglas N.; Boffi, Daniele; Bonizzoni, Francesca
- Numerische Mathematik, Vol. 129, Issue 1
Mixed finite elements for numerical weather prediction
journal, August 2012
- Cotter, C. J.; Shipton, J.
- Journal of Computational Physics, Vol. 231, Issue 21
Energy–enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions
journal, November 2018
- Bauer, W.; Cotter, C. J.
- Journal of Computational Physics, Vol. 373
Firedrake: Automating the Finite Element Method by Composing Abstractions
journal, December 2016
- Rathgeber, Florian; Ham, David A.; Mitchell, Lawrence
- ACM Transactions on Mathematical Software, Vol. 43, Issue 3
LFRic: Meeting the challenges of scalability and performance portability in Weather and Climate models
journal, October 2019
- Adams, S. V.; Ford, R. W.; Hambley, M.
- Journal of Parallel and Distributed Computing, Vol. 132
Raviart–Thomas and Brezzi–Douglas–Marini finite‐element approximations of the shallow‐water equations
journal, July 2008
- Rostand, V.; Le Roux, D. Y.
- International Journal for Numerical Methods in Fluids, Vol. 57, Issue 8
A standard test set for numerical approximations to the shallow water equations in spherical geometry
journal, September 1992
- Williamson, David L.; Drake, John B.; Hack, James J.
- Journal of Computational Physics, Vol. 102, Issue 1