Abstract
Numerical schemes used for computational climate modeling and weather prediction are often of second order accuracy. It is well-known that methods of formal order higher than two offer a significant potential gain in computational efficiency. We here present two classes of high order methods for discretization on the surface of a sphere, first finite difference schemes satisfying the summation-by-parts property on the cube sphere grid, secondly finite volume discretizations on unstructured grids with polygonal cells. Furthermore, we also implement the seventh order accurate weighted essentially non-oscillatory (WENO7) scheme for the cube sphere grid. For the finite difference approximation, we prove a stability estimate, derived from projection boundary conditions. For the finite volume method, we develop the implementational details by working in a local coordinate system at each cell. We apply the schemes to compute advection on a sphere, which is a well established test problem. We compare the performance of the methods with respect to accuracy, computational efficiency, and ability to capture discontinuities.
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References
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
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, 211–224 (1992)
Gustafsson, B.: The convergence rate for difference approximations to mixed initial boundary value problems. Math. Comput. 29(130), 396–406 (1975)
Harten, A., Chakravarthy, S.R.: Multi-dimensional ENO schemes for general geometries. Report 91-76, ICASE (1991)
Hu, C., Shu, C.-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1999)
Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
Langer, T., Belyaev, A., Seidel, H.-P.: Spherical barycentric coordinates. In: Sheffer, A., Polthier, K. (eds.) Eurographics Symposium on Geometry Processing, pp. 81–88 (2006)
Olsson, P.: Summation by parts, projections, and stability. I. Math. Comput. 64, 1035–1065 (1995)
Pärt-Enander, E., Sjögreen, B.: Conservative and non-conservative interpolation between overlapping grids for finite volume solutions of hyperbolic problems. Comput. Fluids 23, 551–574 (1994)
Putman, W.M., Lin, S.-J.: Finite-volume transport on various cube-sphere grids. J. Comput. Phys. 227, 55–78 (2007)
Sjögreen, B., Petersson, N.A.: A fourth order accurate finite difference scheme for the elastic wave equation in second order formulation. J. Sci. Comput. (2011). doi:10.1007/s10915-011-9531-1 LLNL-JRNL-483427
Sjögreen, B., Yee, H.C.: Multiresolution wavelet based adaptive numerical dissipation control for shock-turbulence computation. J. Sci. Comput. 20, 211–255 (2004)
Strand, B.: Summation by parts for finite difference approximations for d/dx. J. Comput. Phys. 110, 47–67 (1994)
Taylor, M., Tribbia, J., Iskandarani, M.: The spectral element method for the shallow water equations on the sphere. J. Comput. Phys. 130, 92–108 (1997)
Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091–3120 (2010)
Zhang, Y.-T., Shu, C.-W.: Third order WENO schemes on three dimensional tetrahedral meshes. Commun. Comput. Phys. 5, 836–848 (2009)
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Sjögreen, B. High Order Finite Difference and Finite Volume Methods for Advection on the Sphere. J Sci Comput 51, 703–732 (2012). https://doi.org/10.1007/s10915-011-9527-x
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DOI: https://doi.org/10.1007/s10915-011-9527-x