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 Nnb → ∞, where N is the total number of particles, h is the smoothing length, and Nnb 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 Nnb fixed. We demonstrate that if Nnb 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 Nnb is increased systematically as the resolution is improved. Using analytic arguments, we derive an optimal compromise scaling for Nnb by requiring that this source of error balance that present in the smoothing procedure. For typical choices of the smoothing kernel, we find Nnb vpropN 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.
The Astrophysical Journal
- Pub Date:
- February 2015
- methods: numerical;
- Astrophysics - Cosmology and Nongalactic Astrophysics
- The revised version accepted in ApJ, with typos corrected and references added