×

zbMATH — the first resource for mathematics

Finite volume transport on various cubed-sphere grids. (English) Zbl 1126.76038
Summary: The performance of a multidimensional finite volume transport scheme is evaluated on the cubed-sphere geometry. Advection tests with prescribed winds are used to evaluate a variety of cubed-sphere projections and grid modifications including the gnomonic and conformal mappings, as well as two numerically generated grids by an elliptic solver and spring dynamics. We explore the impact of grid non-orthogonality on advection tests over the corner singularities of the cubed-sphere grids, using some variations of the transport scheme, including the piecewise parabolic method with alternative monotonicity constraints. The advection tests revealed comparable or better accuracy to those of the original latitudinal-longitudinal grid implementation. It is found that slight deviations from orthogonality on the modified cubed-sphere (quasi-orthogonal) grids do not negatively impact the accuracy. In fact, the more uniform version of the quasi-orthogonal cubed-sphere grids provides better overall accuracy than the most orthogonal (and therefore, much less uniform) conformal grid. It is also shown that a simple non-orthogonal extension to the transport equation enables the use of the highly non-orthogonal and computationally more efficient gnomonic grid with acceptable accuracy.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76R99 Diffusion and convection
Software:
chammp
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lin, S.-J., A “vertically lagrangian” finite-volume dynamical core for global models, Monthly weather review, 132, 2293-2307, (2004)
[2] Lin, S.-J.; Atlas, R.; Yeh, K.-S., Global weather prediction and high-end computing at NASA, Computing in science and engineering, 6, 1, 29-35, (2004)
[3] R. Atlas, O. Oreste, B.-W. Shen, S.-J. Lin, J.-D. Chern, W. Putman, T. Lee, K.-S. Yeh, M. Bosilovich, J. Radakovich, Hurricane forecasting with the high-resolution NASA finite-volume general circulation model, Geophysical Research Letters 32 (L03807) (2006).
[4] B.-W. Shen, R. Atlas, J.-D. Chern, O. Reste, S.-J. Lin, T. Lee, J. Chang, The 0.125 degree finite-volume general circulation model on the NASA columbia supercomputer: Preliminary simulations of mesoscale vortices, Geophysical Research Letters 33 (L05801) (2006).
[5] Delworth, T.L., GFDL’s CM2 global coupled climate models – part I: formulation and simulation characteristics, Journal of climate, 19, 5, 643-674, (2006)
[6] Collins, W.D.; Rasch, P.J.; Boville, B.A.; Hack, J.J.; McCaa, J.R.; Williamson, D.L.; Briegleb, B.; Bitz, C.; Lin, S.-J.; Zhang, M., The formulation and atmospheric simulation of the community atmosphere model: CAM3, Journal of climate, 19, 11, 2144-2161, (2006)
[7] Rasch, P.J.; Coleman, D.B.; Mahowald, N.; Williamson, D.L.; Lin, S.-J.; Boville, B.A.; Hess, P., Characteristics of atmospheric transport using three numerical formulations for atmospheric dynamics in a single GCM framework, Journal of climate, 19, 11, 2243-2266, (2006)
[8] Putman, W.; Lin, S.-J.; Shen, B.-W., Cross-platform performance of a portable communications module the nasa finite volume general circulation model, International journal of high performance computing applications, 19, 3, 213-223, (2005)
[9] R. Oehmke, Q. Stout, Parallel adaptive blocks on a sphere, in: 11th SIAM Conference on Parallel Processing for Scientific Computing, 2001.
[10] M. Herzog, C. Jablonowski, R.C. Oehmke, J.E. Penner, Q.F. Stout, B. van Leer, Adaptive grids in climate modeling: Concept and first results, Eos Trans. AGU, 84(46), Fall Meet. Suppl., Abstract A11D-01, 2003.
[11] C. Jablonowski, Adaptive grids in weather and climate modeling, Ph.D. thesis, University of Michigan (2004).
[12] Sadourny, R., Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids, Monthly weather review, 144, 136-144, (1972)
[13] Ronchi, C.; Iacono, R.; Paolucci, P., The “cubed-sphere:” a new method for the solution of partial differential equations in spherical geometry, Journal of computational physics, 124, 93-114, (1996) · Zbl 0849.76049
[14] Rancic, M.; Purser, J.; Messinger, F., A global shallow water model using an expanded spherical cube: gnomonic versus conformal coordinates, Quarterly journal of the royal meteorological society, 122, 959-982, (1996)
[15] Purser, J.; Rancic, M., Smooth quasi-homogeneous gridding of the sphere, Quarterly journal of the royal meteorological society, 124, 637-647, (1998)
[16] Lin, S.-J.; Rood, R., Multidimensional flux form semi-Lagrangian transport schemes, Monthly weather review, 124, 2046-2070, (1996)
[17] Sadourny, R.; Arakawa, A.; Mintz, Y., Integration of the nondivergent barotropic vorticity equation with an icosahedal-hexagonal grid for the sphere, Monthly weather review, 96, 351-356, (1968)
[18] Williamson, D., Integration of the barotropic vorticity equation on a spherical geodesic grid, Tellus, 20, 642-653, (1968)
[19] Ringler, T.; Heikes, R.; Randall, D., Modeling the atmospheric general circulation using a spherical geodesic grid: A new class of dynamical cores, Monthly weather review, 128, 2471-2490, (2000)
[20] Khamayseh, A.; Mastin, C., Surface grid generation based on elliptic PDE models, Applied mathematics and computation, 65, 253-264, (1994) · Zbl 0811.65105
[21] Tomita, H.; Tsugawa, M.; Satoh, M.; Goto, K., Shallow water model on a modified icosahedral geodesic grid by using spring dynamics, Journal of computational physics, 174, 579-613, (2001) · Zbl 1056.76058
[22] Lauritzen, P., A stability analysis of finite-volume advection schemes permitting long time steps, Monthly weather review, 135, 7, 2658-2673, (2007)
[23] Skamarock, W.C., Positive-definite and montonic limiters for unrestricted-timestep transport schemes, Monthly weather review, 134, 2241-2250, (2006)
[24] H. Huynh, Schemes and constraints for advection, in: Fifth International Conference on Numerical Methods in Fluid Dynamics, 1996. · Zbl 0865.62083
[25] Khamayseh, A.; Mastin, C., Computational conformal mapping for surface grid generation, Journal of computational physics, 123, 394-401, (1996) · Zbl 0851.65079
[26] Adcroft, A.; Campin, J.-M.; Hill, C.; Marshall, J., Implementation of an atmosphere – ocean general circulation model on the expanded spherical cube, Monthly weather review, 132, 2845-2863, (2004)
[27] Williamson, D.; Drake, J.; Hack, J.; Jakob, R.; Swarztrauber, P., A standard test set for numerical approximations to the shallow water equations in spherical geometry, Journal of computational physics, 102, 211-224, (1992) · Zbl 0756.76060
[28] Nair, R.; Cote, J.; Staniforth, A., Cascade interpolation for semi-Lagrangian advection over the sphere, Quarterly journal of the royal meteorological society, 125, 1445-1468, (1999)
[29] Nair, R.; Machenhauer, B., The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere, Monthly weather review, 130, 649-667, (2002)
[30] Nair, R.; Thomas, S.; Loft, R., A discontinuous Galerkin transport scheme on the cubed sphere, Monthly weather review, 133, 814-828, (2005)
[31] R. Nair, C. Jablonowski, Moving vortices on the sphere: A test case for horizontal advection problems, Monthly Weather Review, in press.
[32] Zerroukat, M.; Wood, N.; Staniforth, A., A monotonic and positive definite filter for a semi-Lagrangian inherently conserving and efficient (SLICE) scheme, Quarterly journal of the royal meteorological society, 131, 2923-2936, (2005)
[33] Collela, P.; Woodward, P., The piecewise parabolic method (PPM) for gasdynamical simulations, Journal of computational physics, 54, 174-201, (1984) · Zbl 0531.76082
[34] Suresh, A.; Huynh, H.T., Accurate monotonicity-preserving schemes with runge – kutta time stepping, Journal of computational physics, 136, 83-99, (1997) · Zbl 0886.65099
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.