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 N _ { H } 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 N _ { H } , 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 N _ { H } and hence better numerical convergence ( note that computational costs rise sub-linear with N _ { H } ) .
Our analysis also shows that at low N _ { H } the quartic spline kernel with N _ { H } \approx 60 obtains much better convergence then the standard cubic spline .