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

We prove convergence of the Hamiltonian Particle-Mesh (HPM) method, initially proposed by J. Frank, G. Gottwald, and S. Reich, on a pe- riodic domain when applied to the irrotational shallow water equations as a prototypical model for barotropic compressible uid ow. Under appropriate assumptions, most notably suciently fast decay in Fourier space of the global smoothing operator, and a Strang{Fix condition of order 3 for the local par- tition of unity kernel, the HPM method converges as the number of particles tends to innity and the global interaction scale tends to zero in such a way that the average number of particles per computational mesh cell remains con- stant and the number of particles within the global interaction scale tends to innity. The classical SPH method emerges as a particular limiting case of the HPM algorithm and we nd 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 su- perlinear in the number of particles, we can interpret the HPM method as a fast SPH algorithm.