V. Molchanov and M. Oliver,
Convergence of the Hamiltonian
particle-mesh method for barotropic fluid flow,
Math. Comp. 82 (2013), 861-891.
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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