×

zbMATH — the first resource for mathematics

On vector field reconstructions for semi-Lagrangian transport methods on geodesic staggered grids. (English) Zbl 1351.86005
Summary: We analyse several vector reconstruction methods, based on the knowledge of only specific pointwise vector components, and extend their use to non-structured polygonal C-grids on the sphere. The emphasis is on the reconstruction of the vector field at arbitrary locations on the sphere, as required by semi-Lagrangian transport schemes. This is done by first reconstructing the vector field to fixed locations, followed by interpolations with generalized barycentric coordinates. We derive a hybrid scheme, combining the efficiency of Perot’s method with the accuracy of a least square scheme. This method is second order accurate, and has shown to be competitive and computationally efficient. We analysed the vector reconstruction methods within a semi-Lagrangian transport method, and demonstrated that second order accurate reconstructions are enough to fulfil the requirements for second order accurate semi-Lagrangian methods on icosahedral C-grids.

MSC:
86-08 Computational methods for problems pertaining to geophysics
86A10 Meteorology and atmospheric physics
76M12 Finite volume methods applied to problems in fluid mechanics
52B99 Polytopes and polyhedra
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
PDF BibTeX Cite
Full Text: DOI
References:
[1] Skamarock, W. C.; Klemp, J. B.; Duda, M. G.; Fowler, L. D.; Park, S.-H.; Ringler, T. D., A multiscale nonhydrostatic atmospheric model using centroidal Voronoi tessellations and C-grid staggering, Mon. Weather Rev., 140, 3090-3105, (2012)
[2] Wan, H.; Giorgetta, M. A.; Zängl, G.; Restelli, M.; Majewski, D.; Bonaventura, L.; Fröhlich, K.; Reinert, D.; Rípodas, P.; Kornblueh, L.; Förstner, J., The icon-1.2 hydrostatic atmospheric dynamical core on triangular grids part 1: formulation and performance of the baseline version, Geosci. Model Dev., 6, 3, 735-763, (2013)
[3] Staniforth, A.; Thuburn, J., Horizontal grids for global weather and climate prediction models: a review, Q. J. R. Meteorol. Soc., 138, 662, 1-26, (2012)
[4] Ju, L.; Ringler, T. D.; Gunzburger, M., Voronoi tessellations and their application to climate and global modeling, (Lauritzen, P.; Jablonowski, C.; Taylor, M.; Nair, R., Numerical Techniques for Global Atmospheric Models, Lect. Notes Comput. Sci. Eng., vol. 80, (2011), Springer Berlin, Heidelberg), 313-342
[5] Wang, B.; Zhao, G.; Fringer, O., Reconstruction of vector fields for semi-Lagrangian advection on unstructured, staggered grids, Ocean Model., 40, 1, 52-71, (2011)
[6] Raviart, P.; Thomas, J., A mixed finite element method for 2-nd order elliptic problems, (Galligani, I.; Magenes, E., Mathematical Aspects of Finite Element Methods, Lect. Notes Math., vol. 606, (1977), Springer Berlin/Heidelberg), 292-315
[7] Perot, B., Conservation properties of unstructured staggered mesh schemes, J. Comput. Phys., 159, 1, 58-89, (2000) · Zbl 0972.76068
[8] Vidovic, D., Polynomial reconstruction of staggered unstructured vector fields, Theor. Appl. Mech., 36, 85-99, (2009) · Zbl 1224.41029
[9] Bonaventura, L.; Iske, A.; Miglio, E., Kernel-based vector field reconstruction in computational fluid dynamic models, Int. J. Numer. Methods Fluids, 66, 6, 714-729, (2011) · Zbl 1446.76127
[10] Staniforth, A.; Côté, J., Semi-Lagrangian integration schemes for atmospheric models - a review, Mon. Weather Rev., 119, 9, 2206-2223, (1991)
[11] Whitney, H., Geometric integration theory, (1957), Princeton University Press · Zbl 0083.28204
[12] Weller, H.; Thuburn, J.; Cotter, C. J., Computational modes and grid imprinting on five quasi-uniform spherical C-grids, Mon. Weather Rev., 140, 8, 2734-2755, (2012)
[13] Peixoto, P. S.; Barros, S. R.M., Analysis of grid imprinting on geodesic spherical icosahedral grids, J. Comput. Phys., 237, 61-78, (2013) · Zbl 1286.65118
[14] Fasshauer, G., Positive definite kernels: past, present and future, Dolomit. Res. Notes Approx., 4, 21-63, (2011)
[15] Wachspress, E. L., A rational finite element basis, Math. Sci. Eng., vol. 114, (1975), Academic Press · Zbl 0322.65001
[16] Gillette, A.; Rand, A.; Bajaj, C., Error estimates for generalized barycentric interpolation, Adv. Comput. Math., 37, 417-439, (2012) · Zbl 1259.65013
[17] Iske, A.; Kser, M., Conservative semi-Lagrangian advection on adaptive unstructured meshes, Numer. Methods Partial Differ. Equ., 20, 3, 388-411, (2004) · Zbl 1048.65084
[18] Lauritzen, P. H.; Nair, R. D.; Ullrich, P. A., A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid, J. Comput. Phys., 229, 5, 1401-1424, (2010) · Zbl 1329.65198
[19] Robert, A., A stable numerical integration scheme for the primitive meteorological equations, Atmos.-Ocean, 19, 35-46, (1981)
[20] Nair, R. D.; Lauritzen, P. H., A class of deformational flow test cases for linear transport problems on the sphere, J. Comput. Phys., 229, 23, 8868-8887, (2010) · Zbl 1282.86012
[21] Floater, M., Mean value coordinates, Comput. Aided Geom. Des., 20, 1, 19-27, (2003) · Zbl 1069.65553
[22] Alfeld, P.; Neamtu, M.; Schumaker, L. L., Bernstein-bezier polynomials on spheres and sphere-like surfaces, Comput. Aided Geom. Des., 13, 4, 333-349, (1996) · Zbl 0875.68863
[23] Langer, T.; Belyaev, A.; Seidel, H.-P., Spherical barycentric coordinates, (Proceedings of the Fourth Eurographics Symposium on Geometry Processing, SGP’06, (2006)), 81-88
[24] Du, Q.; Gunzburger, M. D.; Ju, L., Constrained centroidal Voronoi tessellations for surfaces, SIAM J. Sci. Comput., 24, 1488-1506, (2003) · Zbl 1036.65101
[25] Ringler, T. D.; Thuburn, J.; Klemp, J. B.; Skamarock, W. C., A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids, J. Comput. Phys., 229, 9, 3065-3090, (2010) · Zbl 1307.76054
[26] Tomita, H.; Tsugawa, M.; Satoh, M.; Goto, K., Shallow water model on a modified icosahedral geodesic grid by using spring dynamics, J. Comput. Phys., 174, 2, 579-613, (2001) · Zbl 1056.76058
[27] Walters, R.; Hanert, E.; Pietrzak, J.; Roux, D. L., Comparison of unstructured, staggered grid methods for the shallow water equations, Ocean Model., 28, 1-3, 106-117, (2009)
[28] Solin, P., Partial differential equations and the finite element method, Pure Appl. Math., (2005), Wiley Hoboken, NJ
[29] Klausen, R.; Rasmussen, A.; Stephansen, A., Velocity interpolation and streamline tracing on irregular geometries, Comput. Geosci., 16, 261-276, (2012) · Zbl 1254.65035
[30] Renka, R. J., Interpolation of data on the surface of a sphere, ACM Trans. Math. Softw., 10, 417-436, (1984) · Zbl 0548.65001
[31] Lawson, C. L.; Hanson, R. J., Solving least squares problems, Ser. Autom. Comput., (1974), Prentice-Hall · Zbl 0860.65028
[32] Buhmann, M. D., Radial basis functions, (2003), Cambridge University Press New York, NY, USA · Zbl 1038.41001
[33] Baxter, B. J.C.; Hubbert, S., Radial basis functions for the sphere, (Recent Progress in Multivariate Approximation, Witten-Bommerholz, 2000, Int. Ser. Numer. Math., vol. 137, (2001), Birkhäuser Basel), 33-47 · Zbl 1035.41012
[34] Hesse, K.; Le Gia, Q. T., Local radial basis function approximation on the sphere, Bull. Aust. Math. Soc., 77, 2, 197-224, (2008) · Zbl 1151.41002
[35] Ruppert, T., Vector field reconstruction by radial basis functions, (2007), Technical University Darmstadt, Diploma thesis
[36] Wan, H., Developing and testing a hydrostatic atmospheric dynamical core on triangular grids, (2009), International Max Planck Research School on Earth System Modelling, Max Planck Institute for Meteorology Hamburg, Germany, Ph.D. thesis
[37] Rípodas, P.; Gassmann, A.; Förstner, J.; Majewski, D.; Giorgetta, M.; Korn, P.; Kornblueh, L.; Wan, H.; Zängl, G.; Bonaventura, L.; Heinze, T., Icosahedral shallow water model (ICOSWM): results of shallow water test cases and sensitivity to model parameters, Geosci. Model Dev. Discuss., 2, 1, 581-638, (2009)
[38] Schaback, R., Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math., 3, 3, 251-264, (1995) · Zbl 0861.65007
[39] Rosatti, G.; Cesari, D.; Bonaventura, L., Semi-implicit, semi-Lagrangian modelling for environmental problems on staggered Cartesian grids with cut cells, J. Comput. Phys., 204, 1, 353-377, (2005) · Zbl 1143.76484
[40] Narcowich, F. J.; Ward, J. D., Generalized Hermite interpolation via matrix-valued conditionally positive definite functions, Math. Comput., 63, 208, 661-687, (1994) · Zbl 0806.41003
[41] Fornberg, B.; Zuev, J., The Runge phenomenon and spatially variable shape parameters in RBF interpolation, Comput. Math. Appl., 54, 3, 379-398, (2007) · Zbl 1128.41001
[42] Fasshauer, G.; McCourt, M., Stable evaluation of Gaussian radial basis function interpolants, SIAM J. Sci. Comput., 34, 2, A737-A762, (2012) · Zbl 1252.65028
[43] Fornberg, B.; Piret, C., A stable algorithm for flat radial basis functions on a sphere, SIAM J. Sci. Comput., 30, 1, 60-80, (2008) · Zbl 1159.65307
[44] Miura, H.; Kimoto, M., A comparison of grid quality of optimized spherical hexagonal-pentagonal geodesic grids, Mon. Weather Rev., 133, 10, 2817-2833, (2005)
[45] Williamson, D. L.; Drake, J. B.; Hack, J. J.; Jakob, R.; Swarztrauber, P. N., A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys., 102, 1, 211-224, (1992) · Zbl 0756.76060
[46] McGregor, J. L., Economical determination of departure points for semi-Lagrangian models, Mon. Weather Rev., 121, 221-230, (1993)
[47] Ritchie, H., Semi-Lagrangian advection on a Gaussian grid, Mon. Weather Rev., 115, 2, 608-619, (1987)
[48] Shepard, D., A two-dimensional interpolation function for irregularly-spaced data, (Proceedings of the 1968 23rd ACM National Conference, ACM’68, (1968), ACM New York, NY, USA), 517-524
[49] Franke, R.; Nielson, G., Smooth interpolation of large sets of scattered data, Int. J. Numer. Methods Eng., 15, 11, 1691-1704, (1980) · Zbl 0444.65011
[50] Renka, R. J., Multivariate interpolation of large sets of scattered data, ACM Trans. Math. Softw., 14, 2, 139-148, (1988) · Zbl 0642.65006
[51] Renka, R. J., Algorithm 660: QSHEP2D: quadratic Shepard method for bivariate interpolation of scattered data, ACM Trans. Math. Softw., 14, 2, 149-150, (1988) · Zbl 0709.65504
[52] Renka, R. J., Algorithm 661: QSHEP3D: quadratic Shepard method for trivariate interpolation of scattered data, ACM Trans. Math. Softw., 14, 2, 151-152, (1988) · Zbl 0709.65502
[53] Farin, G., Surfaces over Dirichlet tessellations, Comput. Aided Geom. Des., 7, 1-4, 281-292, (1990) · Zbl 0728.65013
[54] Hiyoshi, H.; Sugihara, K., Improving the global continuity of the natural neighbor interpolation, (Lagan, A.; Gavrilova, M. L.; Kumar, V.; Mun, Y.; Tan, C. J.K.; Gervasi, O., ICCSA, vol. 3, Lect. Notes Comput. Sci., vol. 3045, (2004), Springer), 71-80 · Zbl 1116.68626
[55] Renka, R. J., Algorithm 773: SSRFPACK: interpolation of scattered data on the surface of a sphere with a surface under tension, ACM Trans. Math. Softw., 23, 435-442, (1997) · Zbl 0903.65006
[56] Thuburn, J.; Ringler, T. D.; Skamarock, W. C.; Klemp, J. B., Numerical representation of geostrophic modes on arbitrarily structured C-grids, J. Comput. Phys., 228, 8321-8335, (2009) · Zbl 1173.86304
[57] McDonald, A., Accuracy of multiply-upstream semi-Lagrangian advective schemes II, Mon. Weather Rev., 115, 7, 1446-1450, (1987)
[58] Durran, D. R., Numerical methods for fluid dynamics: with applications to geophysics, Texts Appl. Math., vol. 32, (2010), Springer · Zbl 1214.76001
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.