Elementary dispersion analysis of some mimetic discretizations on triangular Cgrids
Abstract
Spurious modes supported by triangular Cgrids limit their application for modeling largescale atmospheric and oceanic flows. Their behavior can be modified within a mimetic approach that generalizes the scalar product underlying the triangular Cgrid discretization. The mimetic approach provides a discrete continuity equation which operates on an averaged combination of normal edge velocities instead of normal edge velocities proper. An elementary analysis of the wave dispersion of the new discretization for Poincaré, Rossby and Kelvin waves shows that, although spurious Poincaré modes are preserved, their frequency tends to zero in the limit of small wavenumbers, which removes the divergence noise in this limit. However, the frequencies of spurious and physical modes become close on shorter scales indicating that spurious modes can be excited unless highfrequency shortscale motions are effectively filtered in numerical codes. We argue that filtering by viscous dissipation is more efficient in the mimetic approach than in the standard Cgrid discretization. Lumping of mass matrices appearing with the velocity time derivative in the mimetic discretization only slightly reduces the accuracy of the wave dispersion and can be used in practice. Thus, the mimetic approach cures some difficulties of the traditional triangular Cgrid discretization but may still need appropriatelymore »
 Authors:
 Max Planck Institute for Meteorology, Hamburg (Germany)
 Alfred Wegener Institute for Polar and Marine Research, Bremerhaven (Germany)
 (Russian Federation)
 Publication Date:
 OSTI Identifier:
 22622244
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 330; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ACCURACY; ATMOSPHERES; COMPUTERIZED SIMULATION; CONTINUITY EQUATIONS; FILTERS; GRIDS; MASS; MATHEMATICAL SOLUTIONS; NOISE; VELOCITY; VISCOSITY; WAVE PROPAGATION
Citation Formats
Korn, P., Email: peter.korn@mpimet.mpg.de, Danilov, S., and A.M. Obukhov Institute of Atmospheric Physics, Moscow. Elementary dispersion analysis of some mimetic discretizations on triangular Cgrids. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2016.10.059.
Korn, P., Email: peter.korn@mpimet.mpg.de, Danilov, S., & A.M. Obukhov Institute of Atmospheric Physics, Moscow. Elementary dispersion analysis of some mimetic discretizations on triangular Cgrids. United States. doi:10.1016/J.JCP.2016.10.059.
Korn, P., Email: peter.korn@mpimet.mpg.de, Danilov, S., and A.M. Obukhov Institute of Atmospheric Physics, Moscow. Wed .
"Elementary dispersion analysis of some mimetic discretizations on triangular Cgrids". United States.
doi:10.1016/J.JCP.2016.10.059.
@article{osti_22622244,
title = {Elementary dispersion analysis of some mimetic discretizations on triangular Cgrids},
author = {Korn, P., Email: peter.korn@mpimet.mpg.de and Danilov, S. and A.M. Obukhov Institute of Atmospheric Physics, Moscow},
abstractNote = {Spurious modes supported by triangular Cgrids limit their application for modeling largescale atmospheric and oceanic flows. Their behavior can be modified within a mimetic approach that generalizes the scalar product underlying the triangular Cgrid discretization. The mimetic approach provides a discrete continuity equation which operates on an averaged combination of normal edge velocities instead of normal edge velocities proper. An elementary analysis of the wave dispersion of the new discretization for Poincaré, Rossby and Kelvin waves shows that, although spurious Poincaré modes are preserved, their frequency tends to zero in the limit of small wavenumbers, which removes the divergence noise in this limit. However, the frequencies of spurious and physical modes become close on shorter scales indicating that spurious modes can be excited unless highfrequency shortscale motions are effectively filtered in numerical codes. We argue that filtering by viscous dissipation is more efficient in the mimetic approach than in the standard Cgrid discretization. Lumping of mass matrices appearing with the velocity time derivative in the mimetic discretization only slightly reduces the accuracy of the wave dispersion and can be used in practice. Thus, the mimetic approach cures some difficulties of the traditional triangular Cgrid discretization but may still need appropriately tuned viscosity to filter small scales and high frequencies in solutions of full primitive equations when these are excited by nonlinear dynamics.},
doi = {10.1016/J.JCP.2016.10.059},
journal = {Journal of Computational Physics},
number = ,
volume = 330,
place = {United States},
year = {Wed Feb 01 00:00:00 EST 2017},
month = {Wed Feb 01 00:00:00 EST 2017}
}

The authors have constructed reliable finite difference methods for approximating the solution to Maxwell`s equations using accurate discrete analogs of differential operators that satisfy the identifies and theorems of vector and tensor calculus in discrete form. The numerical approximation does not have spurious modes and mimics many fundamental properties of the underlying physical problem including conservation laws, symmetries in the solution, and the nondivergence of particular vector fields. Numerical examples demonstrate the high quality of the method when the medium is strongly discontinuous and for nonorthogonal, nonsmooth computational grids.

Finitedifference schemes on regular triangular grids
The phase error and isotropy properties of various finitedifference schemes on grids consisting of regular triangles are compared with similar schemes on square grids. The comparisons are based on a Fourier analysis of semidiscrete solutions to the twodimensional linear convection equation. The finitedifference schemes presented on the triangular grid include a secondorder method, a compact fourthorder method, and a modified compact method designed to extend the accurate wave number range of the numerical approximation. All of the schemes considered are centered and hence nondissipative. In each case, the finitedifference scheme on the triangular grid reduces the anisotropy of the phasemore »