A new insight into the consistency of the SPH interpolation formula

In this paper, the consistency of the smoothed particle hydrodynamics (SPH) interpolation formula is investigated by analytical means. A novel error analysis is developed in n-dimensional space using the Poisson summation formula, which enables the simultaneous treatment of both the kernel and particle approximation errors for arbitrary particle distributions. New consistency integral relations are derived for the particle approximation, which correspond to the cosine Fourier transform of the kernel consistency conditions. The functional dependence of the error bounds on the SPH interpolation parameters, namely the smoothing length, h, and the number of particles within the kernel support, N, is demonstrated explicitly from which consistency conditions arise. As N→∞, the particle approximation converges to the kernel approximation independently of h provided that the particle mass scales with h as m∝hβ with β > n, where n is the spatial dimension. This implies that as h → 0, the joint limit m → 0, N→∞, and N → ∞ is necessary for complete convergence to the continuum, where N is the total number of particles. The analysis also reveals a dominant error term of the form (lnN)n/N for finite N, as it has long been conjectured based on the similarity between the SPH and the quasi-Monte Carlo estimates. When N≫1, the error of the SPH interpolant decays as N−1 independently of the dimension. This ensures approximate partition of unity of the kernel volume.