Numerical Convergence in Smoothed Particle Hydrodynamics

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 $N\to \infty$, $h\to 0$, and $N_{nb}\to \infty$, 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 $N\to \infty$ and $h\to 0$ 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 $N\to \infty$ and $h\to 0$. 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 $N_{nb}\propto N^{1/2}$. 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 $O(N^{1+\delta })$, where $\delta \approx 1/2$, with a weak dependence on the form of the smoothing kernel.

The revised version accepted in ApJ, with typos corrected and references added

References in corpus (5)

• The EAGLE project: Simulating the evolution and assembly of galaxies and their environments
• Introducing the Illustris Project: Simulating the coevolution of dark and visible matter in the Universe
• Properties of galaxies reproduced by a hydrodynamic simulation
• Modelling discontinuities and Kelvin-Helmholtz instabilities in SPH
• Inviscid SPH

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• Frequency of the dark matter subhalo collisions and bifurcation sequence arising formation of dwarf galaxies
• Gas Cooling in Hydrodynamic Simulations with An Exact Time Integration Scheme
• Smoothed Particle Hydrodynamics Physically Reconsidered -- The Relation to Explicit Large Eddy Simulation and the Issue of Particle Duality
• Novel Hydrodynamic Schemes Capturing Shocks and Contact Discontinuities and Comparison Study with Existing Methods
• Mixing Sinc kernels to improve interpolations in smoothed particle hydrodynamics without pairing instability
• Evidence on the incompatibility of smoothed particle hydrodynamics and eddy viscosity models for large eddy simulations
• SPH modeling of natural convection around a heated horizontal cylinder: A comparison with experiments