A Hamiltonian Particle-Mesh Method for the Rotating Shallow-Water Equations

  • Jason Frank
  • Georg Gottwald
  • Sebastian Reich
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 26)

Abstract

A new particle-mesh method is proposed for the rotating shallow-water equations. The spatially truncated equations are Hamiltonian and satisfy a Kelvin circulation theorem. The generation of non-smooth components in the layer-depth is avoided by applying a smoothing operator similar to what has recently been discussed in the context of α-Euler models.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978.Google Scholar
  2. 2.
    C. de Boor, A practical guide to splines, Springer-Verlag, 1978.Google Scholar
  3. 3.
    J. Brackbill & H. Ruppel, Flip: a method for adaptively zoned particle in cell calculations of fluid flows in two dimensions, J. Comput. Phys. 65, 1986, 314–343.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    D.G. Dritschel, L.M. Polvani, & A.R. Mohebalhojeh, The contour-advective semi-Lagrangian algorithm for the shallow water equations, Mon. Weather Rev. 127, 1999, 1551–1565.CrossRefGoogle Scholar
  5. 5.
    D.R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Springer-Ver lag, New York, 1999.Google Scholar
  6. 6.
    J.E. Frank & S. Reich, Conservation properties of smoothed particle hydrodynamics applied to the shallow water equations, submitted.Google Scholar
  7. 7.
    Ch. Gauger, P. Leinen, & H. Yserentant, The finite mass method, SIAM  J. Numer. Anal. 37, 1768–1799, 2000.MathSciNetMATHGoogle Scholar
  8. 8.
    F.H. Harlow, The particle-in-cell computing methods for fluid dynamics, Methods Comput. Phys. 3, 1964, 319–343.Google Scholar
  9. 9.
    R. W. Hockney & J. W. Eastwood, Computer Simulation Using Particles, Adam Hilger, Bristol, New York, 1988.MATHCrossRefGoogle Scholar
  10. 10.
    D. D. Holm, Fluctuation efects on 3D Lagrangian mean and Eulerian mean fluid motion, Physica D 133, 1999, 215–269.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    R.I. McLachlan, On the numerical integration of ODEs by symmetric composition methods, SIAM  J. Sci. Comput. 16, 1995, 151–168.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    A. R. Mohebalhojeh & D. G. Dritschel, On the representation of gravity waves in numerical models of the shallow-water equations, Q. J. R. Meteorol. Soc. 126, 2000, 669–688.CrossRefGoogle Scholar
  13. 13.
    J. J. Monaghan, On the problem of penetration in particle methods, J. Comput. Phys. 82, 1989, 1–15.MATHCrossRefGoogle Scholar
  14. 14.
    J. J. Monaghan, Smoothed particle hydrodynamics, Ann. Rev. Astron. Astrophys. 30 (1992), 543–574.CrossRefGoogle Scholar
  15. 15.
    R. Salmon, Practical use of Hamilton’s principle, J. Fluid Mech. 132, 431–444, 1983.MATHCrossRefGoogle Scholar
  16. 16.
    R. Salmon, Lectures on Geophysical Fluid Dynamics, Oxford University Press, Oxford, 1999.Google Scholar
  17. 17.
    J.M. Sanz-Serna & M.P. Calvo, Numerical Hamiltonian Problems, Chapman & Hall, London, 1994.MATHGoogle Scholar
  18. 18.
    H. Yserentant, A new class of particle methods., Numer. Math. 76, 1997, 87–109.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Jason Frank
    • 1
  • Georg Gottwald
    • 2
  • Sebastian Reich
    • 3
  1. 1.CWIAmsterdamThe Netherlands
  2. 2.Department of MathematicsUniversity of SurreyGuildfordUK
  3. 3.Department of MathematicsImperial CollegeLondonUK

Personalised recommendations