• Arakawa, A., 1966: Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I. J. Comput. Phys., 1 , 119143.

    • Search Google Scholar
    • Export Citation
  • Arnold, V., 1989: Mathematical Methods of Classical Mechanics. 2nd ed. Springer-Verlag, 509 pp.

  • Bridges, T. J., , and S. Reich, 2006: Numerical methods for Hamiltonian PDEs. J. Phys. Math. Gen., 39 (19) 52875320.

  • Dubinkina, S., , and J. Frank, 2007: Statistical mechanics of Arakawa’s discretizations. J. Comput. Phys., 227 , 12861305.

  • Durran, D., 1998: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer-Verlag, 482 pp.

  • Egger, J., 1996: Volume conservation in phase space: A fresh look at numerical integration schemes. Mon. Wea. Rev., 124 , 19551964.

  • Ehrendorfer, M., 1994a: The Liouville equation and its potential usefulness for the prediction of forecast skill. Part I: Theory. Mon. Wea. Rev., 122 , 703713.

    • Search Google Scholar
    • Export Citation
  • Ehrendorfer, M., 1994b: The Liouville equation and its potential usefulness for the prediction of forecast skill. Part II: Applications. Mon. Wea. Rev., 122 , 714728.

    • Search Google Scholar
    • Export Citation
  • Evensen, G., 2006: Data Assimilation: The Ensemble Kalman Filter. Springer-Verlag, 280 pp.

  • Frank, J., , and S. Reich, 2004: The Hamiltonian particle-mesh method for the spherical shallow water equations. Atmos. Sci. Lett., 5 , 8995.

    • Search Google Scholar
    • Export Citation
  • Galewsky, J., , R. Scott, , and L. Polvani, 2004: An initial-value problem for testing numerical models of the global shallow-water equations. Tellus, 56A , 429440.

    • Search Google Scholar
    • Export Citation
  • Hairer, E., , C. Lubich, , and G. Wanner, 2006: Geometric Numerical Integration. 2nd ed. Springer, 515 pp.

  • Hundertmark, T., , and S. Reich, 2007: A regularization approach for a vertical-slice model and semi-Lagrangian Störmer-Verlet time stepping. Quart. J. Roy. Meteor. Soc., 133 , 15751587.

    • Search Google Scholar
    • Export Citation
  • Lamb, J. S. W., 1996: Area-preserving dynamics that is not reversible. Physica A: Stat. Mech. Appl., 228 , (1–4). 344365.

  • Leimkuhler, B., , and S. Reich, 2004: Simulating Hamiltonian Dynamics. Cambridge University Press, 379 pp.

  • Morrison, P., 1998: Hamiltonian description of the ideal fluid. Rev. Mod. Phys., 70 , 467521.

  • Posch, H., , and W. Hoover, 2004: Large-system phase space dimensionality loss in stationary heat flows. Physica D, 187 , 281293.

  • Reich, S., 1999: Backward error analysis for numerical integrators. SIAM J. Numer. Anal., 36 , 475491.

  • Reich, S., 2006: Linearly implicit time stepping methods for numerical weather prediction. BIT Numer. Math., 46 , 607616.

  • Salmon, R., 1983: Practical use of Hamilton’s principle. J. Fluid Mech., 132 , 431444.

  • Salmon, R., 1999: Lectures on Geophysical Fluid Dynamics. Oxford University Press, 400 pp.

  • Salmon, R., 2005: A general method for conserving quantities related to potential vorticity in numerical models. Nonlinearity, 18 , R1R16.

    • Search Google Scholar
    • Export Citation
  • Shepherd, T., 1990: Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Advances in Geophysics, Vol. 32, Academic Press, 287–338.

    • Search Google Scholar
    • Export Citation
  • Sommer, M., , and P. Névir, 2009: A conservative scheme for the shallow-water system on a staggered geodesic grid based on a Nambu representation. Quart. J. Roy. Meteor. Soc., 135 , 485494.

    • Search Google Scholar
    • Export Citation
  • Staniforth, A., , N. Wood, , and S. Reich, 2006: A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations. Quart. J. Roy. Meteor. Soc., 132 , 31073116.

    • Search Google Scholar
    • Export Citation
  • Vichnevetsky, R., , and J. Bowles, 1982: Fourier Analysis of Numerical Approximations of Hyperbolic Equations. SIAM, 152 pp.

  • Williamson, D. L., , J. B. Drake, , J. J. Hack, , R. Jakob, , and P. N. Swarztrauber, 1992: A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys., 102 , 211224.

    • Search Google Scholar
    • Export Citation
  • Zeitlin, V., 1991: Finite-mode analogs of 2D ideal hydrodynamcis: Coadjoint orbits and local canonical structure. Physica D, 49 , 353362.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 137 137 44
PDF Downloads 97 97 40

Phase Space Volume Conservation under Space and Time Discretization Schemes for the Shallow-Water Equations

View More View Less
  • 1 Universität Wien, Fakultät für Mathematik, Wien, Austria
  • 2 Universität Potsdam, Institut für Mathematik, Potsdam, Germany
© Get Permissions
Restricted access

Abstract

Applying concepts of analytical mechanics to numerical discretization techniques for geophysical flows has recently been proposed. So far, mostly the role of the conservation laws for energy- and vorticity-based quantities has been discussed, but recently the conservation of phase space volume has also been addressed. This topic relates directly to questions in statistical fluid mechanics and in ensemble weather and climate forecasting. Here, phase space volume behavior of different spatial and temporal discretization schemes for the shallow-water equations on the sphere are investigated. Combinations of spatially symmetric and common temporal discretizations are compared. Furthermore, the relation between time reversibility and long-time volume averages is addressed.

Corresponding author address: Matthias Sommer, Universität Wien, Fakultät für Mathematik, Nordbergstraße 15, 1090 Wien, Austria. Email: matthias.sommer@univie.ac.at

Abstract

Applying concepts of analytical mechanics to numerical discretization techniques for geophysical flows has recently been proposed. So far, mostly the role of the conservation laws for energy- and vorticity-based quantities has been discussed, but recently the conservation of phase space volume has also been addressed. This topic relates directly to questions in statistical fluid mechanics and in ensemble weather and climate forecasting. Here, phase space volume behavior of different spatial and temporal discretization schemes for the shallow-water equations on the sphere are investigated. Combinations of spatially symmetric and common temporal discretizations are compared. Furthermore, the relation between time reversibility and long-time volume averages is addressed.

Corresponding author address: Matthias Sommer, Universität Wien, Fakultät für Mathematik, Nordbergstraße 15, 1090 Wien, Austria. Email: matthias.sommer@univie.ac.at

Save