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Title: 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]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  2. Eindhoven Univ. of Technology (Netherlands). Dept. of Mechanical Engineering
  3. Delft Univ. of Technology (Netherlands). Faculty of Aerospace Engineering
Publication Date:
Report Number(s):
LA-UR-17-24044
Journal ID: ISSN 0021-9991
Grant/Contract Number:
AC52-06NA25396
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 ; 2017-05-16 - 2017-05-19 ; Boulder, Colorado, United States; Journal ID: ISSN 0021-9991
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) (SC-23)
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}
}