Numerical convergence in smoothed particle hydrodynamics

We study the convergence properties of smoothed particle hydrodynamics (SPH) using numerical tests and simple analytic considerations. Our analysis shows that formal numerical convergence is possible in SPH only in the joint limit N → ∞, h → 0, and N_nb → ∞, where N is the total number of particles, h is the smoothing length, and N_nb is the number of neighbor particles within the smoothing volume used to compute smoothed estimates. Previous work has generally assumed that the conditions N → ∞ and h → 0 are sufficient to achieve convergence, while holding N_nb fixed. We demonstrate that if N_nb is held fixed as the resolution is increased, there will be a residual source of error that does not vanish as N → ∞ and h → 0. Formal numerical convergence in SPH is possible only if N_nb is increased systematically as the resolution is improved. Using analytic arguments, we derive an optimal compromise scaling for N_nb by requiring that this source of error balance that present in the smoothing procedure. For typical choices of the smoothing kernel, we find N_nb ∝N ^0.5. This means that if SPH is to be used as a numerically convergent method, the required computational cost does not scale with particle number as O(N), but rather as O(N ^{1 + δ}), where δ ≈ 0.5, with a weak dependence on the form of the smoothing kernel.