  We study the impact of different discretization choices on the accuracy of
SPH and we explore them in a large number of Newtonian and special-relativistic
benchmark tests. As a first improvement, we explore a gradient prescription
that requires the (analytical) inversion of a small matrix. For a regular
particle distribution this improves gradient accuracies by approximately ten
orders of magnitude and the SPH formulations with this gradient outperform the
standard approach in all benchmark tests. Second, we demonstrate that a simple
change of the kernel function can substantially increase the accuracy of an SPH
scheme. While the "standard" cubic spline kernel generally performs poorly, the
best overall performance is found for a high-order Wendland kernel which allows
for only very little velocity noise and enforces a very regular particle
distribution, even in highly dynamical tests. Third, we explore new SPH volume
elements that enhance the treatment of fluid instabilities and, last, but not
least, we design new dissipation triggers. They switch on near shocks and in
regions where the flow --without dissipation-- starts to become noisy. The
resulting new SPH formulation yields excellent results even in challenging
tests where standard techniques fail completely.
