  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.
