Symmetry group of the equiangular cubed sphere
Author:
Jean-Baptiste Bellet
Journal:
Quart. Appl. Math. 80 (2022), 69-86
MSC (2020):
Primary 86A08, 86A10, 65M50, 20B30, 52B15
DOI:
https://doi.org/10.1090/qam/1604
Published electronically:
November 19, 2021
MathSciNet review:
4360550
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The equiangular cubed sphere is a spherical grid, widely used in computational physics. This paper deals with mathematical properties of this grid. We identify the symmetry group, i.e. the group of the orthogonal transformations that leave the cubed sphere invariant. The main result is that it coincides with the symmetry group of a cube. The proposed proof emphasizes metric properties of the cubed sphere. We study the geodesic distance on the grid, which reveals that the shortest geodesic arcs match with the vertices of a cuboctahedron. The results of this paper lay the foundation for future numerical schemes, based on rotational invariance of the cubed sphere.
References
- M. A. Armstrong, Groups and symmetry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1988. MR 965514, DOI 10.1007/978-1-4757-4034-9
- J.-B. Bellet, M. Brachet, and J.-P. Croisille, Quadrature and symmetry on the cubed sphere, working paper, https://hal.archives-ouvertes.fr/hal-03223150, 2021.
- M. Brachet, Schémas compacts hermitiens sur la Sphère: applications en climatologie et océanographie numérique, PhD thesis, 2018 (in French).
- M. Brachet and J.-P. Croisille, Spherical shallow water simulation by a cubed sphere finite difference solver, Quarterly Journal of the Royal Meteorological Society 147(2021), no. 735, 786–800.
- Chungang Chen and Feng Xiao, Shallow water model on cubed-sphere by multi-moment finite volume method, J. Comput. Phys. 227 (2008), no. 10, 5019–5044. MR 2414846, DOI 10.1016/j.jcp.2008.01.033
- S. Chevrot, R. Martin, and D. Komatitsch, Optimized discrete wavelet transforms in the cubed sphere with the lifting scheme–implications for global finite-frequency tomography, Geophysical Journal International 191(2012), no. 3, 1391–1402.
- M. Faham and H. Nasir, Weakly orthogonal spherical harmonics in a non-polar spherical coordinates and its application to functions on cubed-sphere, Sultan Qaboos University Journal for Science 17(2012), no. 2, 200–213.
- Lucian Ivan, Hans De Sterck, Scott A. Northrup, and Clinton P. T. Groth, Multi-dimensional finite-volume scheme for hyperbolic conservation laws on three-dimensional solution-adaptive cubed-sphere grids, J. Comput. Phys. 255 (2013), 205–227. MR 3109785, DOI 10.1016/j.jcp.2013.08.008
- B. A. Jones, G. H. Born, and G. Beylkin, Comparison of the cubed-sphere gravity model with the spherical harmonics, Journal of Guidance, Control, and Dynamics 33(2010), no. 2, 415–425.
- Hyun-Gyu Kang and Hyeong-Bin Cheong, An efficient implementation of a high-order filter for a cubed-sphere spectral element model, J. Comput. Phys. 332 (2017), 66–82. MR 3591171, DOI 10.1016/j.jcp.2016.12.001
- Y. Kosmann-Schwarzbach, Groupes et symétries : groupes finis, groupes et algèbres de Lie, représentations, Éditions de l’École Polytechnique, 2006.
- D. Lee and A. Palha, A mixed mimetic spectral element model of the rotating shallow water equations on the cubed sphere, J. Comput. Phys. 375 (2018), 240–262. MR 3874535, DOI 10.1016/j.jcp.2018.08.042
- J. L. McGregor, Semi-Lagrangian advection on conformal-cubic grids, Monthly Weather Review 124(1996), no. 6, 1311–1322.
- R. D. Nair, S. J. Thomas, and R. D. Loft, A discontinuous Galerkin transport scheme on the cubed sphere, Monthly Weather Review 133(2005), no. 4, 814–828.
- G. Peyré, L’algèbre discrète de la transformée de Fourier, Ellipses, 2004.
- Brice Portelenelle and Jean-Pierre Croisille, An efficient quadrature rule on the cubed sphere, J. Comput. Appl. Math. 328 (2018), 59–74. MR 3697089, DOI 10.1016/j.cam.2017.06.027
- R. J. Purser, Sets of optimally diversified polyhedral orientations, Office note, National Centers for Environmental Prediction (U.S.), 489, 2017.
- R. J. Purser, Möbius net cubed-sphere gnomonic grids, Office note, National Meteorological Center (U.S.), 496, 2018.
- R. J. Purser and M. Rančić, Smooth quasi-homogeneous gridding of the sphere, Quarterly Journal of the Royal Meteorological Society 124(1998), no. 546, 637–647.
- R. J. Purser and M. Tong, A minor modification of the gnomonic cubed-shaped sphere grid that offers advantages in the context of implementing moving hurricane nests, Office note, National Centers for Environmental Prediction (U.S.), 486, 2017.
- W. M. Putman, Development of the finite-volume dynamical core on the cubed-sphere, PhD thesis, 2007.
- M. Rančić, R. J. Purser, and F. Mesinger, A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates, Quarterly Journal of the Royal Meteorological Society 122(1996), no. 532, 959–982.
- M. Rančić, R. J. Purser, D. Jović, R. Vasic, and T. Black, A nonhydrostatic multiscale model on the uniform Jacobian cubed sphere, Monthly Weather Review 145(2017), no. 3, 1083–1105.
- Erick Rodriguez Bazan and Evelyne Hubert, Multivariate interpolation: preserving and exploiting symmetry, J. Symbolic Comput. 107 (2021), 1–22. MR 4215093, DOI 10.1016/j.jsc.2021.01.004
- C. Ronchi, R. Iacono, and P. S. Paolucci, The “cubed sphere”: a new method for the solution of partial differential equations in spherical geometry, J. Comput. Phys. 124 (1996), no. 1, 93–114. MR 1378553, DOI 10.1006/jcph.1996.0047
- James A. Rossmanith, A wave propagation method for hyperbolic systems on the sphere, J. Comput. Phys. 213 (2006), no. 2, 629–658. MR 2208375, DOI 10.1016/j.jcp.2005.08.027
- R. Sadourny, Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids, Monthly Weather Review 100(1972), no. 2, 136–144.
- S. L. Sobolev, Cubature formulas on the sphere which are invariant under transformations of finite rotation groups, Dokl. Akad. Nauk SSSR 146 (1962), 310–313 (Russian). MR 0141225
- Stephen J. Thomas, John M. Dennis, Henry M. Tufo, and Paul F. Fischer, A Schwarz preconditioner for the cubed-sphere, SIAM J. Sci. Comput. 25 (2003), no. 2, 442–453. MR 2058069, DOI 10.1137/S1064827502409420
- Paul Aaron Ullrich, Peter Hjort Lauritzen, and Christiane Jablonowski, Some considerations for high-order ‘incremental remap’-based transport schemes: edges, reconstructions, and area integration, Internat. J. Numer. Methods Fluids 71 (2013), no. 9, 1131–1151. MR 3023649, DOI 10.1002/fld.3703
- Warren M. Washington, Lawrence Buja, and Anthony Craig, The computational future for climate and Earth system models: on the path to petaflop and beyond, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 (2009), no. 1890, 833–846. MR 2471742, DOI 10.1098/rsta.2008.0219
- D. L. Williamson, The evolution of dynamical cores for global atmospheric models, Journal of the Meteorological Society of Japan. Ser. II 85B(2007), 241–269.
References
- M. A. Armstrong, Groups and symmetry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1988. MR 965514, DOI 10.1007/978-1-4757-4034-9
- J.-B. Bellet, M. Brachet, and J.-P. Croisille, Quadrature and symmetry on the cubed sphere, working paper, https://hal.archives-ouvertes.fr/hal-03223150, 2021.
- M. Brachet, Schémas compacts hermitiens sur la Sphère: applications en climatologie et océanographie numérique, PhD thesis, 2018 (in French).
- M. Brachet and J.-P. Croisille, Spherical shallow water simulation by a cubed sphere finite difference solver, Quarterly Journal of the Royal Meteorological Society 147(2021), no. 735, 786–800.
- Chungang Chen and Feng Xiao, Shallow water model on cubed-sphere by multi-moment finite volume method, J. Comput. Phys. 227 (2008), no. 10, 5019–5044. MR 2414846, DOI 10.1016/j.jcp.2008.01.033
- S. Chevrot, R. Martin, and D. Komatitsch, Optimized discrete wavelet transforms in the cubed sphere with the lifting scheme–implications for global finite-frequency tomography, Geophysical Journal International 191(2012), no. 3, 1391–1402.
- M. Faham and H. Nasir, Weakly orthogonal spherical harmonics in a non-polar spherical coordinates and its application to functions on cubed-sphere, Sultan Qaboos University Journal for Science 17(2012), no. 2, 200–213.
- Lucian Ivan, Hans De Sterck, Scott A. Northrup, and Clinton P. T. Groth, Multi-dimensional finite-volume scheme for hyperbolic conservation laws on three-dimensional solution-adaptive cubed-sphere grids, J. Comput. Phys. 255 (2013), 205–227. MR 3109785, DOI 10.1016/j.jcp.2013.08.008
- B. A. Jones, G. H. Born, and G. Beylkin, Comparison of the cubed-sphere gravity model with the spherical harmonics, Journal of Guidance, Control, and Dynamics 33(2010), no. 2, 415–425.
- Hyun-Gyu Kang and Hyeong-Bin Cheong, An efficient implementation of a high-order filter for a cubed-sphere spectral element model, J. Comput. Phys. 332 (2017), 66–82. MR 3591171, DOI 10.1016/j.jcp.2016.12.001
- Y. Kosmann-Schwarzbach, Groupes et symétries : groupes finis, groupes et algèbres de Lie, représentations, Éditions de l’École Polytechnique, 2006.
- D. Lee and A. Palha, A mixed mimetic spectral element model of the rotating shallow water equations on the cubed sphere, J. Comput. Phys. 375 (2018), 240–262. MR 3874535, DOI 10.1016/j.jcp.2018.08.042
- J. L. McGregor, Semi-Lagrangian advection on conformal-cubic grids, Monthly Weather Review 124(1996), no. 6, 1311–1322.
- R. D. Nair, S. J. Thomas, and R. D. Loft, A discontinuous Galerkin transport scheme on the cubed sphere, Monthly Weather Review 133(2005), no. 4, 814–828.
- G. Peyré, L’algèbre discrète de la transformée de Fourier, Ellipses, 2004.
- Brice Portelenelle and Jean-Pierre Croisille, An efficient quadrature rule on the cubed sphere, J. Comput. Appl. Math. 328 (2018), 59–74. MR 3697089, DOI 10.1016/j.cam.2017.06.027
- R. J. Purser, Sets of optimally diversified polyhedral orientations, Office note, National Centers for Environmental Prediction (U.S.), 489, 2017.
- R. J. Purser, Möbius net cubed-sphere gnomonic grids, Office note, National Meteorological Center (U.S.), 496, 2018.
- R. J. Purser and M. Rančić, Smooth quasi-homogeneous gridding of the sphere, Quarterly Journal of the Royal Meteorological Society 124(1998), no. 546, 637–647.
- R. J. Purser and M. Tong, A minor modification of the gnomonic cubed-shaped sphere grid that offers advantages in the context of implementing moving hurricane nests, Office note, National Centers for Environmental Prediction (U.S.), 486, 2017.
- W. M. Putman, Development of the finite-volume dynamical core on the cubed-sphere, PhD thesis, 2007.
- M. Rančić, R. J. Purser, and F. Mesinger, A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates, Quarterly Journal of the Royal Meteorological Society 122(1996), no. 532, 959–982.
- M. Rančić, R. J. Purser, D. Jović, R. Vasic, and T. Black, A nonhydrostatic multiscale model on the uniform Jacobian cubed sphere, Monthly Weather Review 145(2017), no. 3, 1083–1105.
- Erick Rodriguez Bazan and Evelyne Hubert, Multivariate interpolation: preserving and exploiting symmetry, J. Symbolic Comput. 107 (2021), 1–22. MR 4215093, DOI 10.1016/j.jsc.2021.01.004
- C. Ronchi, R. Iacono, and P. S. Paolucci, The “cubed sphere”: a new method for the solution of partial differential equations in spherical geometry, J. Comput. Phys. 124 (1996), no. 1, 93–114. MR 1378553, DOI 10.1006/jcph.1996.0047
- James A. Rossmanith, A wave propagation method for hyperbolic systems on the sphere, J. Comput. Phys. 213 (2006), no. 2, 629–658. MR 2208375, DOI 10.1016/j.jcp.2005.08.027
- R. Sadourny, Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids, Monthly Weather Review 100(1972), no. 2, 136–144.
- S. L. Sobolev, Cubature formulas on the sphere which are invariant under transformations of finite rotation groups, Dokl. Akad. Nauk SSSR 146 (1962), 310–313 (Russian). MR 0141225
- Stephen J. Thomas, John M. Dennis, Henry M. Tufo, and Paul F. Fischer, A Schwarz preconditioner for the cubed-sphere, SIAM J. Sci. Comput. 25 (2003), no. 2, 442–453. MR 2058069, DOI 10.1137/S1064827502409420
- Paul Aaron Ullrich, Peter Hjort Lauritzen, and Christiane Jablonowski, Some considerations for high-order ‘incremental remap’-based transport schemes: edges, reconstructions, and area integration, Internat. J. Numer. Methods Fluids 71 (2013), no. 9, 1131–1151. MR 3023649, DOI 10.1002/fld.3703
- Warren M. Washington, Lawrence Buja, and Anthony Craig, The computational future for climate and Earth system models: on the path to petaflop and beyond, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 (2009), no. 1890, 833–846. MR 2471742, DOI 10.1098/rsta.2008.0219
- D. L. Williamson, The evolution of dynamical cores for global atmospheric models, Journal of the Meteorological Society of Japan. Ser. II 85B(2007), 241–269.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2020):
86A08,
86A10,
65M50,
20B30,
52B15
Retrieve articles in all journals
with MSC (2020):
86A08,
86A10,
65M50,
20B30,
52B15
Additional Information
Jean-Baptiste Bellet
Affiliation:
Université de Lorraine, CNRS, IECL, F-57000 Metz, France
MR Author ID:
925061
Email:
jean-baptiste.bellet@univ-lorraine.fr
Keywords:
Cubed sphere,
spherical grid,
symmetry,
octahedral group
Received by editor(s):
May 27, 2021
Received by editor(s) in revised form:
October 4, 2021
Published electronically:
November 19, 2021
Article copyright:
© Copyright 2021
Brown University
