Discrete conservation properties for shallow water flows using mixed mimetic spectral elements
Here, a mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element histopolation functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension. These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as higher moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exact spatial integration. The method possesses a weak form statement of geostrophic balance due to the compatible nature of the solution spaces and arbitrarily high order spatial error convergence.
 Authors:

^{[1]};
^{[2]};
^{[3]}
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Eindhoven Univ. of Technology (Netherlands). Dept. of Mechanical Engineering
 Delft Univ. of Technology (Netherlands). Faculty of Aerospace Engineering
 Publication Date:
 Report Number(s):
 LAUR1724044
Journal ID: ISSN 00219991
 Grant/Contract Number:
 AC5206NA25396
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 357; Journal Issue: C; Conference: IMAGe 2017 Theme of the Year: Workshop on Multiscale Geoscience Numerics ; 20170516  20170519 ; Boulder, Colorado, United States; Journal ID: ISSN 00219991
 Publisher:
 Elsevier
 Research Org:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org:
 USDOE Office of Science (SC). Biological and Environmental Research (BER) (SC23)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; 58 GEOSCIENCES; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mimetic; Spectral elements; High order; Shallow water; Energy and Potential enstrophy conservation
 OSTI Identifier:
 1467260
Lee, David Robert, Palha, Artur, and Gerritsma, Marc. Discrete conservation properties for shallow water flows using mixed mimetic spectral elements. United States: N. p.,
Web. doi:10.1016/j.jcp.2017.12.022.
Lee, David Robert, Palha, Artur, & Gerritsma, Marc. Discrete conservation properties for shallow water flows using mixed mimetic spectral elements. United States. doi:10.1016/j.jcp.2017.12.022.
Lee, David Robert, Palha, Artur, and Gerritsma, Marc. 2017.
"Discrete conservation properties for shallow water flows using mixed mimetic spectral elements". United States.
doi:10.1016/j.jcp.2017.12.022. https://www.osti.gov/servlets/purl/1467260.
@article{osti_1467260,
title = {Discrete conservation properties for shallow water flows using mixed mimetic spectral elements},
author = {Lee, David Robert and Palha, Artur and Gerritsma, Marc},
abstractNote = {Here, a mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element histopolation functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension. These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as higher moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exact spatial integration. The method possesses a weak form statement of geostrophic balance due to the compatible nature of the solution spaces and arbitrarily high order spatial error convergence.},
doi = {10.1016/j.jcp.2017.12.022},
journal = {Journal of Computational Physics},
number = C,
volume = 357,
place = {United States},
year = {2017},
month = {12}
}