Skip to main content Accessibility help
×
Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-25T13:50:39.503Z Has data issue: false hasContentIssue false

7 - Numerical Methods for Advection

Published online by Cambridge University Press:  15 May 2017

Guy P. Brasseur
Affiliation:
Max-Planck-Institut für Meteorologie, Hamburg
Daniel J. Jacob
Affiliation:
Harvard University, Massachusetts
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allen, D. J., Douglass, A. R., Rood, R. B., and Guthrie, P. D. (1991) Application of a monotonic upstream-biased transport scheme to three-dimensional constituent transport calculations, Mon. Wea. Rev., 119, 24562464.Google Scholar
Asselin, R. (1972) Frequency filter for time integrations, Mon. Wea. Rev., 100, 487490.Google Scholar
Bermejo, R. (1990) Notes and correspondence on the equivalence of semi-Lagrangian schemes and particle-in-cell finite element methods, Mon. Wea. Rev., 118, 979987.Google Scholar
Boris, J. P. and Book, D. L. (1973) Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works, J. Comput. Phys., 11, 3869.Google Scholar
Boris, J. P. and Book, D. L. (1976) Flux-corrected transport. III. Minimal-error FCT algorithms, J. Comput. Phys., 20, 397431.CrossRefGoogle Scholar
Bott, A. (1989a) A positive definite advection scheme obtained by non-linear renormalization of the advective fluxes, Mon. Wea. Rev., 117, 10061015.2.0.CO;2>CrossRefGoogle Scholar
Bott, A. (1989b) Reply, Mon. Wea. Rev., 117, 26332636.Google Scholar
Carpenter, R. L., Droegemeier, K. K., Woodward, P. R., and Hane, C. E. (1990) Application of the piecewise parabolic method (PPM) to meteorological modeling. Mon. Wea. Rev., 118, 586612.2.0.CO;2>CrossRefGoogle Scholar
Chen, Y. and Falconer, R. A. (1992) Advection–diffusion modelling using the modified QUICK scheme, Int. J. Numer. Meth. Fluids, 15, 11711196.CrossRefGoogle Scholar
Chlond, A. (1994) Locally modified version of Bott’s advection scheme, Mon. Wea. Rev., 122, 111125.2.0.CO;2>CrossRefGoogle Scholar
Colella, P. and Sekora, M. D. (2008) A limiter for PPM that preserves accuracy at smooth extrema, J. Comput. Phys., 227 (15), 70697076.Google Scholar
Colella, P. and Woodward, P. R. (1984) The piecewise parabolic method (PPM) for gasdynamical simulations, J. Comput. Phys., 54, 174201.Google Scholar
Courant, R., Friedrichs, K., and Lewy, H. (1928) Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann., 100, 3274.CrossRefGoogle Scholar
Courant, R., Isaacson, E., and Rees, M. (1952) On the solution of nonlinear hyperbolic differential equations by finite difference, Commun. Pure Appl. Math., 5, 243255.CrossRefGoogle Scholar
Crowley, W. P. (1968) Numerical advection experiments, Mon. Wea. Rev., 96, 111.2.0.CO;2>CrossRefGoogle Scholar
Davies, H. C. (1983) Limitations of some common lateral boundary schemes used in regional NWP models, Mon. Wea. Rev., 111, 10021012.2.0.CO;2>CrossRefGoogle Scholar
Dukowicz, J. K. and Baumgardner, J. R. (2000) Incremental remapping as a transport/advection algorithm, J. Comput. Phys., 160, 318335.Google Scholar
Dullemond, C. P. (2009) Numerical Fluid Dynamics, Lecture Notes, Zentrum für Astronomie, Ruprecht-Karls Universität, Heidelberg.Google Scholar
Farrow, D. E. and Stevens, D. P. (1994) A new tracer advection scheme for Bryan Cox type ocean general circulation models, J. Phys. Oceanogr., 25, 17311741.Google Scholar
Forester, C. K. (1977) Higher order monotonic convective difference schemes, J. Comput. Phys., 23, 122.CrossRefGoogle Scholar
Godunov, S. K. (1959) A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47, 357393.Google Scholar
Gross, E. S., Koseff, J. R., and Monismith, S. G. (1999) Evaluation of advective schemes for estuarine salinity simulations, J. Hydr. Engrg., ASCE, 125 (1) 3246.Google Scholar
Horowitz, L. W., Walters, S., Mauzerall, D. L., et al. (2003) A global simulation of tropospheric ozone and related tracers: Description and evaluation of MOZART, version 2, J. Geophys. Res., 108 (D24), 4784, doi: 10.1029/2002JD002853.Google Scholar
Iserles, A. (1986) Generalised leapfrog methods, IMA J. Numer. Anal., 6, 381392.Google Scholar
Jablonowski, C. and Williamson, D. L. (2011) The pros and cons of diffusion, filters and fixers in atmospheric general circulation models, In Numerical Techniques for Global Atmospheric Models, Lecture Notes in Computational Science and Engineering (Lauritzen, P. H., Jablonowski, C., Taylor, M. A., and Nair, R. D., eds.), Springer, Berlin.Google Scholar
Jöckel, P., von Kuhlmann, R., Lawrence, M. G., et al. (2001) On a fundamental problem in implementing flux-form advection schemes for tracer transport in 3-dimensional general circulation and chemistry transport models, Q. J. R. Meteorol. Soc., 127, 10351052.Google Scholar
Kida, H. (1983) General circulation of air parcels and transport characteristics derived from a hemispheric GCM: Part 1. A determination of advective mass flow in the lower stratosphere, J. Meteor. Soc. Japan, 61, 171187.Google Scholar
Kim, C. (2003) Accurate multi-level schemes for advection, Int. J. Numer. Meth. Fluids, 41, 471494, doi: 10.1002/fld.443.Google Scholar
Kurihara, Y. (1965) On the use of implicit and iterative methods for the time integration of the wave equation, Mon. Wea. Rev., 93, 3346.Google Scholar
Lauritzen, P. H., Kaas, E., and Machenhauer, B. (2006) A mass-conservative semi-implicit semi-Lagrangian limited-area shallow-water model on a sphere, Mon. Wea. Rev., 134, 25882606.CrossRefGoogle Scholar
Lauritzen, P. H., Nair, R. D., and Ullrich, P. A. (2010) A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid, J. Comput. Phys., 229 (5), doi: 10.1016/j.jcp.2009.10.036Google Scholar
Lauritzen, P. H., Ullrich, P. A., and Nair, R. D. (2011) Atmospheric transport schemes: Desirable properties and a semi-Lagrangian view on finite-volume discretizations. In Numerical Techniques for Global Atmospheric Models, Lecture Notes in Computational Science and Engineering (Lauritzen, P. H., Jablonowski, C., Taylor, M. A., and Nair, R. D., eds.), Springer, Berlin.Google Scholar
Lax, P. D. (1954) Weak solutions of nonlinear equations and their numerical computation, Comm. Pure Appl. Math., 7, 159193.Google Scholar
Lax, P. D. and Wendroff, B. (1960) Systems of conservation laws, Comm. Pure and Appl. Math., 13, 217237.CrossRefGoogle Scholar
Lax, P. D. and Wendroff, B. (1964) Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math., 17 (3), 281398.Google Scholar
Leith, C. E. (1965) Numerical simulations of the earth’s atmosphere, Meth. Comput. Phys., 4, 128.Google Scholar
Leonard, B. P. (1979) A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comput. Methods in Appl. Mech. Eng., 19, 5998.CrossRefGoogle Scholar
Lin, J. C. (2012) Lagrangian modeling of the atmosphere: An introduction. In Lagrangian Modeling of the Atmosphere (Lin, J., Brunner, D., Gerbig, C., et al., eds.), American Meteorological Union, Washington, DC.Google Scholar
Lin, S. J. and Rood, R. B. (1996) Multidimensional flux-form semi-Lagrangian transport schemes, Mon. Wea. Rev., 124, 20462070.Google Scholar
Lipscomb, W. H. and Ringler, T. D. (2005) An incremental remapping transport scheme on a spherical geodesic grid, Mon. Wea. Rev., 133, 23352350.CrossRefGoogle Scholar
Miura, H. (2007) An upwind-biased conservative advection scheme for spherical hexagonal–pentagonal grids, Mon. Wea. Rev., 135, 40384044.Google Scholar
Müller, R. (1992), The performance of classical versus modern finite-volume advection schemes for atmospheric modeling in a one-dimensional test-bed, Mon. Wea. Rev., 120, 14071415.Google Scholar
Nair, R. D., and Machenhauer, B. (2002) The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere, Mon. Wea. Rev., 130, 647667.2.0.CO;2>CrossRefGoogle Scholar
Nair, R. D., Scroggs, J. S., and Semazzi, F. H. M. (2003) A forward-trajectory global semi-Lagrangian transport scheme, J. Comput. Phys., 193, 275294.Google Scholar
Nair, R. D., Levy, M., and Lauritzen, P. H. (2011) Emerging methods for conservation laws. In Numerical Techniques for Global Atmospheric Models, Lecture Notes in Computational Science and Engineering (Lauritzen, P. H., Jablonowski, C., Taylor, M. A., and Nair, R. D., eds.), Springer, Berlin.Google Scholar
Papayannis, A., Mamouri, R. E., Amiridis, V., et al. (2012) Optical properties and vertical extension of aged ash layers over the Eastern Mediterranean as observed by Raman lidars during the Eyjafjallajökull eruption in May 2010, Atmos. Env., 48, 5665.CrossRefGoogle Scholar
Patera, A. T. (1984) A spectral element method for fluid dynamics: Laminar flow in a channel expansion, J. Comput. Phys., 54, 468488.Google Scholar
Prather, M. J. (1986) Numerical advection by conservation of second order moments, J. Geophys. Res., 91, 66716681.CrossRefGoogle Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (2007) Numerical Recipes: The Art of Scientific Computing, 3rd edition, Cambridge University Press, Cambridge.Google Scholar
Rasch, P. J. and Williamson, D. L. (1990a) Computational aspects of moisture transport in global models of the atmosphere, Quart. J. Roy. Meteor. Soc., 116, 10711090.CrossRefGoogle Scholar
Rasch, P. J. and Williamson, D. L. (1990b) On shape-preserving interpolation and semi-Lagrangian transport, SIAM J. Sci. Stat. Comput., 11, 656687.CrossRefGoogle Scholar
Ritchie, H. (1986) Eliminating the interpolation associated with the semi-Lagrangian scheme, Mon. Wea. Rev., 114, 135146.2.0.CO;2>CrossRefGoogle Scholar
Roe, P. L. (1986) Characteristic-based schemes for the Euler equations, Ann. Rev. Fluid Mech., 18, 337365.Google Scholar
Russell, G. L. and Lerner, J. A. (1981) A new finite-differencing scheme for the tracer transport equation, J. Appl. Meteor., 20, 14831498.Google Scholar
Seibert, P. and Morariu, B. (1991) Improvements of upstream, semi-Lagrangian numerical advection schemes, J. Appl. Meteor., 30, 117125.2.0.CO;2>CrossRefGoogle Scholar
Shapiro, R. (1971) The use of linear filtering as a parameterization of atmospheric motion, J. Atmos. Sci., 28, 523531.Google Scholar
Skamarock, W. and Menchaca, M. (2010) Conservative transport schemes for spherical geodesic grids: High-order reconstruction for forward-in-time schemes, Mon. Wea. Rev., 138, 44974508.CrossRefGoogle Scholar
Slingerland, R. and Kump, L. (2011) Mathematical Modeling of Earth’s Dynamical Systems: A Primer, Princeton University Press, Princeton, NJ.Google Scholar
Smolarkiewicz, P. K. (1983) A simple positive definite advection scheme with small implicit diffusion, Mon. Wea. Rev., 111, 479486.Google Scholar
Smolarkiewicz, P. K. (1984) A fully multidimensional positive definite advection transport algorithm with small implicit diffusion, J. Comput. Phys., 54, 325362.CrossRefGoogle Scholar
Smolarkiewicz, P. K. (2006) Multidimensional positive definite advection transport algorithm: An overview, Int. J. Numer. Meth. Fluids, 50 (10), 11231144.Google Scholar
Spalding, D. B. (1972) A novel finite difference formulation for differential expressions involving both first and second derivatives, Int. J. Numer. Meth. Eng., 4, 551559.CrossRefGoogle Scholar
Staniforth, A. and Côté, J. (1991) Semi-Lagrangian integration schemes for atmospheric models: A review, Mon. Wea. Rev., 119, 22062223.2.0.CO;2>CrossRefGoogle Scholar
Stohl, A. (1998) Computation, accuracy and applications of trajectories: A review and bibliography, Atmos. Environ., 32 (6), 947966.Google Scholar
Stohl, A., Hittenberger, M., and Wotawa, G. (1998) Validation of the Lagrangian particle dispersion model FLEXPART against large scale tracer experiment data, Atmos. Environ., 32, 42454262.Google Scholar
Tremback, C. J., Powell, J., Cotton, W. R., and Pielke, R. A. (1987), The forward-in-time upstream advection scheme: Extension to higher order, Mon. Wea. Rev., 115, 894902.Google Scholar
van Leer, B. (1977) Toward the ultimate conservative difference scheme IV: A new approach to numerical convection, J. Comp. Phys., 23, 276299.CrossRefGoogle Scholar
van Leer, B. (1979) Toward the ultimate conservative difference scheme V: A second order sequel to Godunov’s method, J. Comp. Phys., 32, 101136.Google Scholar
Warming, R. F. and Beam, R. M. (1976) Upwind second-order difference schemes and applications in aerodynamic flows, AIAA Journal, 14 (9), 12411249.CrossRefGoogle Scholar
Williamson, D. L. (1992) Review of the numerical approaches for modeling global transport. In Air Pollution Modelling and its Applications (Dop, H. V. and Kallow, G., eds.), Plenum Press, New York.Google Scholar
Williamson, D. L. and Rasch, P. J. (1989) Two-dimensional semi-Lagrangian transport with shape preserving interpolation, Mon. Wea. Rev., 117, 102129.Google Scholar
Zerroukat, M., Wood, N., and Staniforth, A. (2002) SLICE: A semi-Lagrangian inherently conserving and efficient scheme for transport problems, Q. J. R. Meteorol. Soc., 128, 28012820.CrossRefGoogle Scholar
Zerroukat, M., Wood, N., and Staniforth, A. (2007) Application of the Parabolic Spline Method (PSM) to a multi-dimensional conservative transport scheme (SLICE), J. Comput. Phys., 225, 935948.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×