The two traditional methods for numerical hydrodynamics are grid-based or Smoothed Particle Hydrodynamics (SPH). SPH has the advantage (and occasional disadvantage) that resolution is finer in high density regions. Grid-based methods meanwhile employ either a static grid (usually an inefficient choice) or use Adaptive Mesh Refinement (AMR) to concentrate computational resources where they are required. The distribution of refined cells can thus be tuned to any variable of the simulation.

Much has been written about the differences between SPH and AMR and the traditional view is that SPH requires fewer computational overheads and is Galilean invariant while AMR is fundamentally anisotropic but is better at resolving shocks. Code comparisons typically concentrate AMR resolution elements where there are steep gradients in hydrodynamical quantities. It is however rare to use such criteria for simulations of galactic discs or in cosmological volumes, perhaps due to the impractical nature of refining so much of the volume. Refinement criteria are instead based on the Jeans' length or mass of grid cells.

Using simulations of an idealised galactic disc with an external spiral potential realised both with SPH and AMR, we examine whether Jeans' length refinement criteria are adequate for simulations of molecular cloud formation and whether the "conventional wisdom" that AMR is superior to SPH in terms of shock resolution is generally true or function of refinement criteria. We find systematic
differences in the spiral arm density using each method and suggest using additional criteria to improve shock resolution in grid simulations.