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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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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.