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Convergence of the Hamiltonian particle-mesh method for barotropic fluid flow

Authors: Vladimir Molchanov and Marcel Oliver
Journal: Math. Comp. 82 (2013), 861-891
MSC (2010): Primary 76M28, 65M15, 65D07
Published electronically: August 21, 2012
MathSciNet review: 3008841
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Abstract: We prove convergence of the Hamiltonian Particle-Mesh (HPM) method, initially proposed by J. Frank, G. Gottwald, and S. Reich, on a periodic domain when applied to the irrotational shallow water equations as a prototypical model for barotropic compressible fluid flow. Under appropriate assumptions, most notably sufficiently fast decay in Fourier space of the global smoothing operator, and a Strang-Fix condition of order $ 3$ for the local partition of unity kernel, the HPM method converges as the number of particles tends to infinity and the global interaction scale tends to zero in such a way that the average number of particles per computational mesh cell remains constant and the number of particles within the global interaction scale tends to infinity.

The classical SPH method emerges as a particular limiting case of the HPM algorithm and we find that the respective rates of convergence are comparable under suitable assumptions. Since the computational complexity of bare SPH is algebraically superlinear and the complexity of HPM is logarithmically superlinear in the number of particles, we can interpret the HPM method as a fast SPH algorithm.

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  • 1. O. Bokhove, V. Molchanov, M. Oliver, and B. Peeters, On the rate of the convergence of the Hamiltonian particle-mesh method, submitted for publication.
  • 2. C. de Boor, R.A. Devore, and A. Ron, Approximation from shift-invariant subspaces of $ L\sb 2(\mathbb{R}\sp d)$, Trans. Am. Math. Soc. 341 (1994), 787-806. MR 1195508 (94d:41028)
  • 3. R. Di Lisio, A particle method for a self-gravitating fluid: a convergence result, Math. Method. Appl. Sci. 18 (1995), 1083-1094. MR 1357365 (96i:76104)
  • 4. R. Di Lisio, E. Grenier, and M. Pulvirenti, The convergence of the SPH method, Comput. Math. Appl. 35 (1998), 95-102. MR 1605143 (98i:76063)
  • 5. C.J. Cotter, J. Frank, and S. Reich, Hamiltonian particle-mesh method for two-layer shallow-water equations subject to the rigid-lid approximation, SIAM J. Appl. Dyn. Syst. 3 (2004), 69-83. MR 2067898 (2005a:76120)
  • 6. M.F. Dixon, ``Geometric integrators for continuum dynamics.'' PhD Thesis, Imperial College, 2007.
  • 7. M. Dubal, A. Staniforth, N. Wood, and S. Reich, Analysis of a regularized, time-staggered discretization applied to a vertical slice model, Atmos. Sci. Lett. 7 (2006), 86-92.
  • 8. J. Frank, G. Gottwald, and S. Reich, A Hamiltonian particle-mesh method for the rotating shallow water equations, in: M. Griebel and M.A. Schweitzer, eds., Meshfree Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering 26, Springer-Verlag, 2002, pp. 131-142. MR 2003435
  • 9. J. Frank and S. Reich, Conservation properties of smoothed particle hydrodynamics Applied to the Shallow Water Equations, BIT 43 (2003), 41-55. MR 1981639 (2004c:76112)
  • 10. J. Frank and S. Reich, The Hamiltonian particle-mesh method for the spherical shallow water equations, Atmos. Sci. Let. 5 (2004), 89-95.
  • 11. R.A. Gingold and J.J. Monaghan, Smoothed particle hydrodynamics: theory and application to non-spherical stars, Mon. Not. R. Astron. Soc. 181 (1977), 375-389.
  • 12. E. Hairer, C. Lubich, and G. Wanner, ``Geometric Numerical Integration,'' Springer-Verlag, Berlin, 2002. MR 1904823 (2003f:65203)
  • 13. Y. Katznelson, ``An Introduction to Harmonic Analysis.'' Third edition. Cambridge University Press, Cambridge, 2004. MR 2039503 (2005d:43001)
  • 14. B. Leimkuhler and S. Reich, ``Simulating Hamiltonian Dynamics,'' Cambridge University Press, 2005. MR 2132573 (2006a:37078)
  • 15. S. Li and W.K. Liu, Meshfree and particle methods and their applications, Appl. Mech. Rev. 55 (2002), 1-34.
  • 16. L.B. Lucy, A numerical approach to the testing of the fission hypothesis, Astrophys. J. 82 (1977), 1013.
  • 17. A. Majda, ``Introduction to PDEs and Waves for the Atmosphere and Ocean,'' Courant Institute of Mathematical Sciences, New York and American Mathematical Society, Providence, RI, 2003. MR 1965452 (2004b:76152)
  • 18. V. Molchanov, ``Particle-Mesh and Meshless Methods for a Class of Barotropic Fluids,'' PhD Thesis, Jacobs University, 2008.
  • 19. V. Molchanov and M. Oliver, Convergence of the Hamiltonian particle-mesh method applied to barotropic fluid equations, Proc. Appl. Math. Mech. 8 (2008), 10127-10128.
  • 20. J.J. Monaghan, Smoothed particle hydrodynamics, Annu. Rev. Astron. Astrophys. 30 (1992), 543-574.
  • 21. J.J. Monaghan, Smoothed particle hydrodynamics, Rep. Prog. Phys. 68 (2005), 1703-1759. MR 2158506 (2006g:76093)
  • 22. J.P. Morris, ``Analysis of Smoothed Particle Hydrodynamics with Applications.'' PhD Thesis, Monash University, 1996.
  • 23. K. Oelschläger, On the connection between Hamiltonian many-particle systems and the hydrodynamical equations, Arch. Rational Mech. Anal. 115 (1991), 297-310. MR 1120850 (92h:82087)
  • 24. D.J. Price, ``Magnetic Fields in Astrophysics.'' PhD Thesis, University of Cambridge, 2004.
  • 25. R. Salmon, ``Lectures on Geophysical Fluid Dynamics.'' Oxford University Press, Oxford, 1998. MR 1718369 (2002d:86001)
  • 26. I.J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions. Part A: On the problem of smoothing or graduation. A first class of analytic approximation formulae. Part B: On the problem of osculatory interpolation. A second class of analytic approximation formulae. Q. Appl. Math. 4 (1946), 45-99; 112-141. MR 0015914 (7:487b)
  • 27. E.M. Stein and G. Weiss, ``Introduction to Fourier Analysis on Euclidean Spaces.'' Princeton University Press, Princeton, NJ, 1971. MR 0304972 (46:4102)
  • 28. G. Strang and G. Fix, A Fourier analysis of the finite element variational method, C.I.M.E. II, Ciclo Erice 1971, Constructive Aspects of Functional Analysis (G. Geymonat, ed.), 1973, pp. 793-840.
  • 29. Z. Wu, Compactly supported radial functions and the Strang-Fix condition, Appl. Math. Comput. 84 (1997), 115-124. MR 1449691 (98c:65018)

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Additional Information

Vladimir Molchanov
Affiliation: School of Engineering and Science. Jacobs University, 28759 Bremen, Germany

Marcel Oliver
Affiliation: School of Engineering and Science. Jacobs University, 28759 Bremen, Germany

Received by editor(s): January 6, 2011
Received by editor(s) in revised form: September 21, 2011, and September 28, 2011
Published electronically: August 21, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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