Abstract
Advection schemes are important in many branches of computational fluid dynamics. They are used for tracer transport in atmospheric models, where the tracer mixing ratios must remain positive. Many advection schemes employ monotonic limiters, however these can reduce the accuracy of the schemes for smooth data. In this article a commonly used monotonic limiter is modified to make it positive definite (but not strictly monotonic). Testing in multiple dimensions shows that there is improved accuracy over the monotonic limiter, while no negative values are produced.
References
Chawla, A., Spindler, D.M., Tolman, H.L.: Validation of a thirty year wave hindcast using the Climate Forecast System Reanalysis winds. Ocean Model. 70, 189–206 (2013)
Holdaway, D., Kent, J.: Assessing the tangent linear behavior of common tracer transport schemes and their use in a linearized atmospheric general circulation model. Tellus A 67, 27895 (2015). https://doi.org/10.3402/tellusa.v67.27895
Holdaway, D., Thuburn, J., Wood, N.: On the relation between order of accuracy, convergence rate and spectral slope for linear numerical methods applied to multiscale problems. Int. J. Numer. Methods Fluids 56, 1297–1303 (2008)
Lauritzen, P.H., Thuburn, J.: Evaluating advection/transport schemes using interrelated tracers, scatter plots and numericalmixing diagnostics. Q. J. R. Meteorol. Soc. 138, 906–918 (2012)
Lauritzen, P.H., Ullrich, P.A., Jablonowski, C., Bosler, P.A., Calhoun, D., Conley, A.J., Enomoto, T., Dong, L., Dubey, S., Guba, O., Hansen, A.B., Kaas, E., Kent, J., Lamarque, J.F., Prather, M.J., Reinert, D., Shashkin, V.V., Skamarock, W.C., Sorensen, B., Taylor, M.A., Tolstykh, M.A.: A standard test case suite for two-dimensional linear transport on the sphere: results from a collection of state-of-the-art schemes. Geosci. Model Dev. 5(2012), 887–901 (2014)
Lax, P.D., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13, 217–237 (1960)
Leonard, B.P.: The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection. Comput. Methods Appl. Mech. Eng. 88, 17–74 (1991)
Lin, S.J., Rood, R.B.: Multidimensional flux-form semi-Lagrangian transport schemes. Mon. Weather Rev. 124, 2046–2070 (1996)
Miura, H.: An upwind-biased conservative advection scheme for spherical hexagonal-pentagonal grids. Mon. Weather Rev. 135, 4038–4044 (2007)
Nair, R.D., Lauritzen, P.H.: A class of deformational flow test cases for linear transport problems on the sphere. J. Comput. Phys. 229, 8868–8887 (2010)
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, 3065–3090 (2010)
Smolarkiewicz, P.K.: A simple positive definite advection scheme with small implicit diffusion. Mon. Weather Rev. 111, 479–486 (1983)
Thuburn, J.: Multidimensional flux-limited advection schemes. J. Comput. Phys. 23, 74–83 (1996)
Tremback, C.J., Powell, J., Cotton, W.R., Pielke, R.A.: The forward-in-time upstream advection scheme: extension to higher orders. Mon. Weather Rev. 115, 540–555 (1987)
Zerroukat, M., Wood, N., Staniforth, A.: The parabolic spline method (PSM) for conservative transport problems. Int. J. Numer. Methods Fluids 51, 1297–1318 (2005)
Acknowledgements
I would like to thank two anonymous reviewers for their comments that greatly improved the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kent, J. A Positive Definite Limiter for Advection Problems. J Sci Comput 82, 73 (2020). https://doi.org/10.1007/s10915-020-01178-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-020-01178-0