A Necessary Condition for Individual Time Steps in SPH Simulations

We show that the smoothed particle hydrodynamics (SPH) method, used with individual time-steps in the way described in the literature, cannot handle strong explosion problems correctly. In the individual time-step scheme, particles determine their time-steps essentially from a local Courant condition. Thus they cannot respond to a strong shock, if the pre-shock timescale is too long compared to the shock timescale. This problem is not severe in SPH simulations of galaxy formation with a temperature cutoff in the cooling function at 10⁴ K, while it is very dangerous for simulations in which the multiphase nature of the interstellar medium under 10⁴ K is taken into account. A solution for this problem is to introduce a time-step limiter which reduces the time-step of a particle if it is too long compared to the time-steps of its neighbor particles. Thus this kind of time-step constraint is essential for the correct treatment of explosions in high-resolution SPH simulations with individual time-steps.