Convergence of AMR and SPH simulations - I. Hydrodynamical resolution and convergence tests

We compare the results for a set of hydrodynamical tests performed with the adaptive mesh refinement finite volume code, MG, and the smoothed particle hydrodynamics (SPH) code, SEREN. The test suite includes shock tube tests, with and without cooling, the non-linear thin-shell instability and the Kelvin-Helmholtz instability. The main conclusions are the following. (i) The two methods converge in the limit of high resolution and accuracy in most cases. All tests show good agreement when numerical effects (e. g. discontinuities in SPH) are properly treated. (ii) Both methods can capture adiabatic shocks and well-resolved cooling shocks perfectly well with standard prescriptions. However, they both have problems when dealing with under-resolved cooling shocks, or strictly isothermal shocks, at high Mach numbers. The finite volume code only works well at first order and even then requires some additional artificial viscosity. SPH requires either a larger value of the artificial viscosity parameter, alpha(AV), or a modified form of the standard artificial viscosity term using the harmonic mean of the density, rather than the arithmetic mean. (iii) Some SPH simulations require larger kernels to increase neighbour number and reduce particle noise in order to achieve agreement with finite volume simulations (e. g. the Kelvin-Helmholtz instability). However, this is partly due to the need to reduce noise that can corrupt the growth of small-scale perturbations (e. g. the Kelvin-Helmholtz instability). In contrast, instabilities seeded from large-scale perturbations (e. g. the non-linear thin shell instability) do not require more neighbours and hence work well with standard SPH formulations and converge with the finite volume simulations. (iv) For purely hydrodynamical problems, SPH simulations take an order of magnitude longer to run than finite volume simulations when running at equivalent resolutions, i. e. when they both resolve the underlying physics to the same degree. This requires about two to three times as many particles as the number of cells.