Abstract
The spectral element method is well known as an efficient way to obtain high-order numerical solutions on unstructured finite element grids. However, the oscillatory nature of the method’s advection operator makes it unsuitable for many applications. One popular way to address this problem is with high-order discontinuous-Galerkin methods. In this work, an alternative solution which fits within the continuous Galerkin formulation of the spectral element method is proposed. Making use of a compatible formulation of spectral elements, a natural way to implement conservative non-oscillatory reconstructions for spectral element advection is shown. The reconstructions are local to the element and thus preserve the parallel efficiency of the method. Numerical results from a low-order quasi-monotone reconstruction and a higher-order sign-preserving reconstruction are presented.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Maday, Y., Patera, A.T.: Spectral element methods for the incompressible Navier Stokes equations. In: Noor, A.K., Oden, J.T. (eds.) State of the Art Surveys on Computational Mechanics, pp. 71–143. ASME, New York (1987)
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)
Giraldo, F.X.: A spectral element shallow water model on spherical geodesic grids. International Journal for Numerical Methods in Fluids 35, 869–901 (2001)
Fournier, A., Taylor, M., Tribbia, J.: The spectral element atmosphere model (SEAM): High-resolution parallel computation and localized resolution of regional dynamics. Mon. Wea. Rev. 132, 726–748 (2004)
Thomas, S., Loft, R.: The NCAR spectral element climate dynamical core: Semi-implicit eulerian formulation. J. Sci. Comput. 25, 307–322 (2005)
Dennis, J., Fournier, A., Spotz, W.F., St -Cyr, A., Taylor, M.A., Thomas, S.J., Tufo, H.: High resolution mesh convergence properties and parallel efficiency of a spectral element atmospheric dynamical core. Int. J. High Perf. Comput. Appl. 19, 225–235 (2005)
Wang, H., Tribbia, J.J., Baer, F., Fournier, A., Taylor, M.A.: A spectral element version of CAM2. Monthly Weather Review 135 (2007)
Haidvogel, D., Curchitser, E.N., Iskandarani, M., Hughes, R., Taylor, M.A.: Global modeling of the ocean and atmosphere using the spectral element method. Atmosphere-Ocean Special 35, 505–531 (1997)
Molcard, A., Pinardi, N., Iskandarani, M., Haidvogel, D.: Wind driven circulation of the mediterranean sea simulated with a spectral element ocean model. Dynamics of Atmospheres and Oceans 35, 97–130 (2002)
Komatitsch, D., Tromp, J.: Spectral-element simulations of global seismic wave propagation - I. validation. Geophys. J. Int. 149, 390–412 (2002)
Komatitsch, D., Tsuboi, S., Ji, C., Tromp, J.: A 14.6 billion degrees of freedom, 5 teraflops, 2.5 terabyte earthquake simulation on the earth simulator. In: Proceedings of the ACM / IEEE Supercomputing SC 2003 conference (2003)
Bhanot, G., Dennis, J.M., Edwards, J., Grabowski, W., Gupta, M., Jordan, K., Loft, R.D., Sexton, J., St-Cyr, A., Thomas, S.J., Tufo, H.M., Voran, T., Walkup, R., Wyszogrodski, A.A.: Early experiences with the 360TF IBM BlueGene/L platform. International Journal of Computational Methods 5, 237–253 (2008)
Taylor, M.A., Edwards, J., St-Cyr, A.: Petascale atmospheric models for the community climate system model: New developments and evaluation of scalable dynamical cores. J. Phys. Conf. Ser. 125(012023) (2008)
Iskandarani, M., Levin, J., Choi, B.J., Haidvogel, D.: Comparison of advection schemes for high-order hp finite element and finite volume methods. Ocean Modelling 10, 233–252 (2005)
Rančić, M., Purser, R., Mesinger, F.: A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates. Q. J. R. Meteorol. Soc. 122, 959–982 (1996)
Fournier, A., Rosenberg, D., Pouquet, A.: Dynamically adaptive spectral-element simulations of 2d incompressible navier-stokes vortex decays. Geophysical and Astrophysical Fluid Dynamics (2009) (to appear)
Taylor, M.A., Edwards, J., Thomas, S., Nair, R.: A mass and energy conserving spectral element atmospheric dynamical core on the cubed-sphere grid. J. Phys. Conf. Ser. 78(012074) (2007)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Taylor, M.A., Cyr, A.S., Fournier, A. (2009). A Non-oscillatory Advection Operator for the Compatible Spectral Element Method . In: Allen, G., Nabrzyski, J., Seidel, E., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds) Computational Science – ICCS 2009. ICCS 2009. Lecture Notes in Computer Science, vol 5545. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01973-9_31
Download citation
DOI: https://doi.org/10.1007/978-3-642-01973-9_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-01972-2
Online ISBN: 978-3-642-01973-9
eBook Packages: Computer ScienceComputer Science (R0)