Skip to main content
SpringerLink
Log in

Finite Difference Schemes on Sparse and Full Grids

  • Chapter
  • First Online:
Mathematics of the Weather

Part of the book series: Springer Atmospheric Sciences ((SPRINGERATMO))

  • 308 Accesses

Abstract

The notion of FDs on sparse grids is introduced. It means that from a regular distribution of grid points not all are used, which offers the opportunity to make models more efficient for the same resolution. This section aims at transferring some of the 1D schemes defined in Chapter “Local-Galerkin Schemes in 1D” to two dimensions. There is no way the most general L-Galerkin scheme or a class of such schemes can be presented. The number of possibilities is too large. So just examples are presented to show how the schemes work. In particular, most examples are 2D. While the complexity of computer programs normally increases substantially when going to 3D, it is often obvious how to proceed from 2D to 3D.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Adcroft A, Hill C, Marshall J (1997) Representation of topography by shaved cells in a height coordinate ocean model. Mon Wea Rev 125(9):2293–2315

    Article  Google Scholar 

  • Arakawa A, Lamb VR (1977) Computational design of the basic dynamical processes of the UCLA general circulation model, 1st edn. University of California, Los Angeles, pp. 173–265

    Google Scholar 

  • Baumgardner JR, Frederickson PO (1985) Icosahedral discretization of the two-sphere. SIAM J Numer Anal 22:107–115

    Article  Google Scholar 

  • Braun VG (1958) Schlichting Grenzschicht theorie, 1st edn. Braun, Karlsruhe

    Google Scholar 

  • Cannon F, Carvalho LMV, Jones C, Norris J, Bookhagen B, Kiladis GN (2017) Effects of topographic smoothing on the simulation of winter precipitation in High Mountain Asia. JGR Atmos 122(3):1456–1474

    Article  Google Scholar 

  • Cotter J, Shipton J (2012) Mixed finite elements for numerical weather prediction. J Comp Phys 231(21):7076–7091

    Article  Google Scholar 

  • Davies DC (1976) A lateral boundary formulation for multi-level prediction models. Q J R Meteorol Soc 102:405–418

    Google Scholar 

  • Doms G, Schättler U (2002) A description of the nonhydrostatic regional model LM. Part I: dynamics and numerics. Consortium for small-scale modeling (COSMO) LM F90 2.18, Tech. Rep., DWD, Germany

    Google Scholar 

  • Dudhia J (1993) A nonhydrostatic version of the Penn State-NCAR mesoscale model: validation tests. Mon Wea Rev 121:1439–1513

    Article  Google Scholar 

  • Durran DR (2010) Numerical methods for fluid dynamics: with applications to geophysics, 2nd edn. Springer, New York, pp. 35–146

    Book  Google Scholar 

  • Gallus WA, Klemp JB (2000) Behavior of flow over step orography. Mon Wea Rev 128:1153–1164

    Article  Google Scholar 

  • Giraldo FX, Perotb JB, Fisher PF (2003) A spectral element semi-Lagrangian (SESL) method for the spherical shallow water equations. J Comp Phys 190(2):623–650

    Article  Google Scholar 

  • Gresho PM, Lee RL, Sani RI (1977) Advection-dominated flows with emphasis on the consequence of mass lumping. Finite elements in fluids, vol 3, 1st edn. Wiley, New York, pp. 335–350

    Google Scholar 

  • Herrington AR, Lauritzen PH, Taylor MA, Goldhaber S, Eaton BE, Reed KA, Ullrich PA (2019) Physics-dynamics coupling with element-based high-order Galerkin methods: quasi equal-area physics grid. Mon Wea Rev 147:69–84

    Article  Google Scholar 

  • Melvin T, Benacchio T, Milano P, Shipway BJ, Wood N (2019) A mixed finite-element, finite-volume, semi-implicit discretisation for atmospheric dynamics: Cartesian geometry. Quart J Roy Meteor Soc 145(724):2835–2853

    Article  Google Scholar 

  • Mesinger F (1981) Horizontal advection schemes of a staggered gridâăťan enstrophy and energy-conserving model. Mon Wea Rev 109(3):467–478

    Article  Google Scholar 

  • Pedlosky J (1987) Geophysical fluid dynamics, 1st edn. Springer, New York, pp. 745–792

    Book  Google Scholar 

  • Sadourny R (1972) Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids. Mon Wea Rev 100:136–144

    Article  Google Scholar 

  • Sadourny R, Morel P (1969) A finite-difference approximation of the primitive equations for a hexagonal grid on a plane. Mon Wea Rev 97:439–445

    Article  Google Scholar 

  • Saito K, Doms G (1998) 3-D mountain waves by the Lokal-Modell of DWD and the MRI mesoscale nonhydrostatic model. Meteorol Geophys 49(1):7–19

    Article  Google Scholar 

  • Satoh M, Tomita H, Yashiro H, Miura H, Kodama C, Seiki T, Noda A, Yamada Y, Goto D, Sawada M, Miyoshi T, Niwa Y, Hara M, Ohno T, Iga S, Arakawa T, Inoue T, Kubokawa H (2014) The non-hydrostatic icosahedral atmospheric model: description and development. Progr Earth Planet Sci 18 1. Article number: 18

    Google Scholar 

  • Shaw J, Weller H (2016) Comparison of terrain-following and cut-cell grids using a nonhydrostatic model. Mon Wea Rev 144:2085–2099

    Article  Google Scholar 

  • Steppeler J (1990) Simple test calculations concerning finite element applications to numerical weather prediction. Numer Method Fluids 11:209–226

    Article  Google Scholar 

  • Steppeler J (1993) The southern oscillation as an example of a simple ordered subsystem of a complex chaotic system. J Climate 10:473–480

    Article  Google Scholar 

  • Steppeler J, Klemp JB (2017) Advection on cut-cell grids for an idealized mountain of constant slope. Mon Wea Rev 145:1765–1777

    Article  Google Scholar 

  • Steppeler J, Navon IM, Lu H (1990) Finite element schemes for extended integrations of atmospheric models. J Comput Phys 130:213–235

    Google Scholar 

  • Steppeler J, Bitzer HW, Minotte M, Bonaventura L (2002) Nonhydrostatic atmospheric modeling using a z-coordinate representation. Mon Wea Rev 130:2143–2149

    Article  Google Scholar 

  • Steppeler J, Bitzer HW, Janjic Z, Schättler U, Prohl P, Gjertsen U, Torrisi L, Parfinievicz J, Avgoustoglou E, Damrath U (2006) Prediction of clouds and rain using a z-coordinate nonhydrostatic model. Mon Wea Rev 134:3625–3643

    Article  Google Scholar 

  • Steppeler J, Ripodas P, Thomas S (2008) Third order finite difference schemes on icosahedral-type grids on the sphere. Mon Wea Rev 136:2683–2698

    Article  Google Scholar 

  • Steppeler J, Park SH, Dobler A (2013) Forecasts covering one month using a cut cell model. Geosci Model Dev 6:875–882

    Article  Google Scholar 

  • Steppeler J, Li J, Fang F, Zhu J, Ullrich PA (2019) o3o3: a variant of spectral elements with a regular collocation grid. Mon Wea Rev 147:2067–2082

    Article  Google Scholar 

  • Steppeler J, Li J, Fang F, Navon IM (2019) Third-order sparse grid generalized spectral elements on hexagonal cells for homogeneous advection in a plane. Meteorol Atmos Phys 132(5):703–719

    Article  Google Scholar 

  • Stull RB (2018) Practical meteorology: an algebra-based survey of atmospheric science, 1st edn. The University of British Columbia, Vancouver, pp. 745–792

    Google Scholar 

  • Tomita H, Tsugawa M, Satoh M, Goto K (2001) Shallow water model on a modified icosahedral geodesic grid by using spring dynamics. J Comput Phys 174:579–613

    Article  Google Scholar 

  • Williamson DL (1968) Integrations of the barotropic vorticity equation on a spherical geodesic grid. Tellus 20:643–653

    Article  Google Scholar 

  • Wilson DR, Ballard SP (1999) A microphysically based precipitation scheme for the UK meteorological office unified model. Quart J Roy Meteor Soc 125:1607–1636

    Article  Google Scholar 

  • Yamazaki H, Satomura T (2010) Nonhydrostatic atmospheric modeling using a combined Cartesian grid. Mon Wea Rev 132:3932–3945

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Helmholtz-Zentrum Hereon, Climate Service Center Germany (GERICS), Hamburg, Hamburg, Germany

    Jürgen Steppeler

  2. Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China

    Jinxi Li

Authors
  1. Jürgen Steppeler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinxi Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Steppeler, J., Li, J. (2022). Finite Difference Schemes on Sparse and Full Grids. In: Mathematics of the Weather. Springer Atmospheric Sciences. Springer, Cham. https://doi.org/10.1007/978-3-031-07238-3_5

Download citation

Publish with us

Policies and ethics

Search

Navigation