### Abstract

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.

Original language | English (US) |
---|---|

Article number | 6 |

Journal | Astrophysical Journal Letters |

Volume | 800 |

Issue number | 1 |

DOIs | |

State | Published - Feb 10 2015 |

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### All Science Journal Classification (ASJC) codes

- Astronomy and Astrophysics
- Space and Planetary Science

### Cite this

*Astrophysical Journal Letters*,

*800*(1), [6]. https://doi.org/10.1088/0004-637X/800/1/6

}

*Astrophysical Journal Letters*, vol. 800, no. 1, 6. https://doi.org/10.1088/0004-637X/800/1/6

**Numerical convergence in smoothed particle hydrodynamics.** / Zhu, Qirong; Hernquist, Lars; Li, Yuexing.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Numerical convergence in smoothed particle hydrodynamics

AU - Zhu, Qirong

AU - Hernquist, Lars

AU - Li, Yuexing

PY - 2015/2/10

Y1 - 2015/2/10

N2 - 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 ∝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.

AB - 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 ∝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.

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U2 - 10.1088/0004-637X/800/1/6

DO - 10.1088/0004-637X/800/1/6

M3 - Article

AN - SCOPUS:84922550841

VL - 800

JO - Astrophysical Journal Letters

JF - Astrophysical Journal Letters

SN - 2041-8205

IS - 1

M1 - 6

ER -