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.
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