TY - JOUR
T1 - Discrete conservation properties for shallow water flows using mixed mimetic spectral elements
AU - Lee, D.
AU - Palha, A.
AU - Gerritsma, M.
PY - 2018/3/15
Y1 - 2018/3/15
N2 - 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.
AB - 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.
KW - Energy and potential enstrophy conservation
KW - High order
KW - Mimetic
KW - Shallow water
KW - Spectral elements
UR - http://www.scopus.com/inward/record.url?scp=85039997422&partnerID=8YFLogxK
U2 - 10.1016/j.jcp.2017.12.022
DO - 10.1016/j.jcp.2017.12.022
M3 - Article
AN - SCOPUS:85039997422
SN - 0021-9991
VL - 357
SP - 282
EP - 304
JO - Journal of Computational Physics
JF - Journal of Computational Physics
ER -