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

Molchanov, Vladimir; Oliver, Marcel

doi:10.1090/S0025-5718-2012-02648-2

Convergence of the Hamiltonian particle-mesh method for barotropic fluid flow
HTML articles powered by AMS MathViewer

by Vladimir Molchanov and Marcel Oliver PDF

Math. Comp. 82 (2013), 861-891 Request permission

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.

References

O. Bokhove, V. Molchanov, M. Oliver, and B. Peeters, On the rate of the convergence of the Hamiltonian particle-mesh method, submitted for publication.
Carl de Boor, Ronald A. DeVore, and Amos Ron, Approximation from shift-invariant subspaces of $L_2(\mathbf R^d)$, Trans. Amer. Math. Soc. 341 (1994), no. 2, 787–806. MR 1195508, DOI 10.1090/S0002-9947-1994-1195508-X
Roberto Di Lisio, A particle method for a self-gravitating fluid: a convergence result, Math. Methods Appl. Sci. 18 (1995), no. 13, 1083–1094. MR 1357365, DOI 10.1002/mma.1670181305
R. Di Lisio, E. Grenier, and M. Pulvirenti, The convergence of the SPH method, Comput. Math. Appl. 35 (1998), no. 1-2, 95–102. Simulation methods in kinetic theory. MR 1605143, DOI 10.1016/S0898-1221(97)00260-5
Colin J. Cotter, Jason Frank, and Sebastian Reich, Hamiltonian particle-mesh method for two-layer shallow-water equations subject to the rigid-lid approximation, SIAM J. Appl. Dyn. Syst. 3 (2004), no. 1, 69–83. MR 2067898, DOI 10.1137/030600076
M.F. Dixon, “Geometric integrators for continuum dynamics.” PhD Thesis, Imperial College, 2007.
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.
Jason Frank, Georg Gottwald, and Sebastian Reich, A Hamiltonian particle-mesh method for the rotating shallow-water equations, Meshfree methods for partial differential equations (Bonn, 2001) Lect. Notes Comput. Sci. Eng., vol. 26, Springer, Berlin, 2003, pp. 131–142. MR 2003435, DOI 10.1007/978-3-642-56103-0_{1}0
Jason Frank and Sebastian Reich, Conservation properties of smoothed particle hydrodynamics applied to the shallow water equation, BIT 43 (2003), no. 1, 41–55. MR 1981639, DOI 10.1023/A:1023620100065
J. Frank and S. Reich, The Hamiltonian particle-mesh method for the spherical shallow water equations, Atmos. Sci. Let. 5 (2004), 89–95.
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.
Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric numerical integration, Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2002. Structure-preserving algorithms for ordinary differential equations. MR 1904823, DOI 10.1007/978-3-662-05018-7
Yitzhak Katznelson, An introduction to harmonic analysis, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. MR 2039503, DOI 10.1017/CBO9781139165372
Benedict Leimkuhler and Sebastian Reich, Simulating Hamiltonian dynamics, Cambridge Monographs on Applied and Computational Mathematics, vol. 14, Cambridge University Press, Cambridge, 2004. MR 2132573
S. Li and W.K. Liu, Meshfree and particle methods and their applications, Appl. Mech. Rev. 55 (2002), 1–34.
L.B. Lucy, A numerical approach to the testing of the fission hypothesis, Astrophys. J. 82 (1977), 1013.
Andrew Majda, Introduction to PDEs and waves for the atmosphere and ocean, Courant Lecture Notes in Mathematics, vol. 9, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 1965452, DOI 10.1090/cln/009
V. Molchanov, “Particle-Mesh and Meshless Methods for a Class of Barotropic Fluids,” PhD Thesis, Jacobs University, 2008.
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.
J.J. Monaghan, Smoothed particle hydrodynamics, Annu. Rev. Astron. Astrophys. 30 (1992), 543–574.
J. J. Monaghan, Smoothed particle hydrodynamics, Rep. Progr. Phys. 68 (2005), no. 8, 1703–1759. MR 2158506, DOI 10.1088/0034-4885/68/8/R01
J.P. Morris, “Analysis of Smoothed Particle Hydrodynamics with Applications.” PhD Thesis, Monash University, 1996.
Karl Oelschläger, On the connection between Hamiltonian many-particle systems and the hydrodynamical equations, Arch. Rational Mech. Anal. 115 (1991), no. 4, 297–310. MR 1120850, DOI 10.1007/BF00375277
D.J. Price, “Magnetic Fields in Astrophysics.” PhD Thesis, University of Cambridge, 2004.
Rick Salmon, Lectures on geophysical fluid dynamics, Oxford University Press, New York, 1998. MR 1718369
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, Quart. Appl. Math. 4 (1946), 45–99. MR 15914, DOI 10.1090/S0033-569X-1946-15914-5
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
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.
Zongmin Wu, Compactly supported radial functions and the Strang-Fix condition, Appl. Math. Comput. 84 (1997), no. 2-3, 115–124. MR 1449691, DOI 10.1016/S0096-3003(96)00110-5