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

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.

On the Rate of Convergence of the Hamiltonian Particle-Mesh Method

“…This corresponds to the analytic results in [11], where the error constants may grow exponentially in time as is typical for general trajectory error estimates for evolution equations. Thus, the Hamiltonian aspects of HPM shall not be considered further.…”

“…We find that the rate of convergence is much better than the best available theoretical estimates in [11]. Further, our results indicate that HPM performs best when the number of particles is on the order of the number of grid cells, the HPM global smoothing kernel has fast decay in Fourier space, and the HPM local interpolation kernel is a cubic spline.…”

“…Comparing the HPM method with classical smoothed particle hydrodynamics (SPH) which also possesses a Hamiltonian structure and associated conservation laws [7,9,13], we note that the known convergence results are very similar in the sense that both can be shown to be of order O(N 2−ε ) for any ε > 0 provided the underlying kernel functions satisfy certain technical conditions [11]. On the other hand, one time step of the HPM method can be computed in O(N ) or O(N ln N ) operations.…”

“…It was originally proposed in the context of the shallow water equations by Frank, Gottwald and Reich [6], and tested on a variety of two-dimensional geophysical flow problems [3,4,5]. Moreover, the HPM method was shown to be convergent [10,11] as the number of particles N tends to infinity.…”

“…A priori, these three parameters may be chosen independently. The theory in [11] yields a relation for the scaling of λ and µ as a function of N to achieve a certain rate of convergence; however, it is not known if this relation is optimal. Further, it requires technical conditions on the kernel functions not all of which are met in [6].…”