  Abridged: We simulate a massive galaxy cluster in a LCDM Universe using three
different approaches to solving the equations of non-radiative hydrodynamics:
`classic' Smoothed Particle Hydrodynamics (SPH); a novel SPH with a higher
order dissipation switch (SPHS); and adaptive mesh refinement (AMR). We find
that SPHS and AMR are in excellent agreement, with both forming a well-defined
entropy core that rapidly converges with increasing mass and force resolution.
By contrast, SPH exhibits rather different behaviour. At low redshift, entropy
decreases systematically with decreasing cluster-centric radius, converging on
ever lower central values with increasing resolution. At higher redshift, SPH
is in better agreement with SPHS and AMR but shows much poorer numerical
convergence. We trace these discrepancies to artificial surface tension in SPH
at phase boundaries. At early times, the passage of massive substructures close
to the cluster centre stirs and shocks gas to build an entropy core. At later
times, artificial surface tension causes low entropy gas to sink artificially
to the centre of the cluster.
  We use SPHS to study the contribution of numerical versus physical
dissipation on the entropy core, and argue that numerical dissipation is
required to ensure single-valued fluid quantities in converging flows. However,
provided this dissipation occurs only at the resolution limit, and provided
that it does not propagate errors to larger scales, its effect is benign. There
is no requirement to build `sub-grid' models of unresolved turbulence for
galaxy cluster simulations. We conclude that entropy cores in non-radiative
simulations of galaxy clusters are physical, resulting from entropy generation
in shocked gas during cluster assembly, putting to rest the long-standing
puzzle of cluster entropy cores in AMR simulations versus their apparent
absence in classic SPH simulations.
