Improving convergence in smoothed particle hydrodynamics simulations without pairing instability

TL;DR

This paper demonstrates that Wendland functions as smoothing kernels in SPH simulations prevent pairing instability, enabling larger neighbor counts and improved convergence without sacrificing computational efficiency, especially in shear flows.

Contribution

The study introduces Wendland kernels as a stable alternative to traditional B-splines in SPH, allowing larger neighbor numbers and better convergence, challenging traditional ideas about pairing instability origins.

Findings

Wendland kernels avoid pairing instability for all neighbor counts.

Wendland kernels are computationally more efficient than higher-order B-splines.

Quartic spline kernel with ~60 neighbors outperforms cubic spline in convergence.

Abstract

The numerical convergence of smoothed particle hydrodynamics (SPH) can be severely restricted by random force errors induced by particle disorder, especially in shear flows, which are ubiquitous in astrophysics. The increase in the number NH of neighbours when switching to more extended smoothing kernels at fixed resolution (using an appropriate definition for the SPH resolution scale) is insufficient to combat these errors. Consequently, trading resolution for better convergence is necessary, but for traditional smoothing kernels this option is limited by the pairing (or clumping) instability. Therefore, we investigate the suitability of the Wendland functions as smoothing kernels and compare them with the traditional B-splines. Linear stability analysis in three dimensions and test simulations demonstrate that the Wendland kernels avoid the pairing instability for all NH, despite having vanishing derivative at the origin (disproving traditional ideas about the origin of this instability; instead, we uncover a relation with the kernel Fourier transform and give an explanation in terms of the SPH density estimator). The Wendland kernels are computationally more convenient than the higher-order B-splines, allowing large NH and hence better numerical convergence (note that computational costs rise sub-linear with NH). Our analysis also shows that at low NH the quartic spline kernel with NH ~= 60 obtains much better convergence then the standard cubic spline.