some test calculations of an isolated galaxy, but realistic calculations to be compared with observations need further implementation of, for example, stellar feedback. @2019MNRAS.482.2555H [hereafter HA19] took another approach to solve the evolution of full grain size distribution. They post-processed an isolated-galaxy simulation: they chose some SPH gas particles and derived the evolution of grain size distribution on each particle based on its history of physical conditions (mainly gas density, temperature, and metallicity). They successfully derived the grain size distribution similar to the MRN distribution at ages comparable to the Milky Way ($\gtrsim 3$ Gyr). However, because their calculation was based on post-processing, the results may depend on the frequency of snapshots; that is, we cannot capture the change of physical conditions shorter than the output snapshot interval ($10^{7}$ yr in the above post-processing). Indeed, the processes occurring in dense environments such as coagulation and accretion could have a shorter characteristic time-scales in metal-rich regions (A17). To obtain robust results against such potentially quick processes, we need to solve the evolution of grain size distribution and that of hydrodynamical structures simultaneously.

The goal of this paper is to implement the evolution of full grain size distribution directly in a hydrodynamic 