the number density of hydrogen nuclei, $n_\mathrm{H}$ as $\rho_\mathrm{gas}=\mu m_\mathrm{H}n_\mathrm{H}$ ($\mu =1.4$ is the gas mass per hydrogen, and $m_\mathrm{H}$ is the mass of hydrogen atom). The dust-to-gas ratio, $\mathcal{D}(t)$, is defined as $$\begin{aligned}
\mathcal{D}_\mathrm{tot}(t)\equiv\frac{\rho_\mathrm{d,tot}(t)}{\rho_\mathrm{gas}(t)}.\label{eq:dg}\end{aligned}$$ Note that the evolution of $\rho_\mathrm{gas}$ is calculated by the hydrodynamic simulation for each gas particle.

The time-evolution of $\rho_\mathrm{d}(m,\, t)$ is described by $$\begin{aligned}
\frac{\upartial\rho_\mathrm{d}(m,\, t)}{\upartial t} &=
\left[\frac{\upartial\rho_\mathrm{d}(m,\, t)}{\upartial t}\right]_\mathrm{star}+
\left[\frac{\upartial\rho_\mathrm{d}(m,\, t)}{\upartial t}\right]_\mathrm{sput}
\nonumber\\
&+
\left[\frac{\upartial\rho_\mathrm{d}(m,\, t)}{\upartial t}\right]_\mathrm{acc}+
\left[\frac{\upartial\rho_\mathrm{d}(m,\, t)}{\upartial t}\right]_\mathrm{shat}
\nonumber\\
&+
\left[\frac{\upartial\rho_\mathrm{d}(m,\, t)}{\upartial t}\right]_\mathrm{coag}
+\rho_\mathrm{d}(m,\, t)\frac{\mathrm{d}\ln\rho_\mathrm{gas}}{\mathrm{d}t},
\label{eq:basic}\end{aligned}$$ where the terms with subscripts ‘star’, ‘sput’, ‘acc’, ‘shat’ and ‘coag’ indicate the changing rates of grain mass distribution by stellar dust production, sputtering, accretion, shattering, and coagulation, respectively, and the last term is caused by the change of background gas density. Below we briefly describe each term. We actually solve discrete forms and adopt 32 grid points for the discrete grain size distribution in the range of $3\times 10^{-4} -10\,\micron$.

### Stellar dust production {#subsec:stellar}

We estimate the increase of dust mass by stellar dust production, assuming a constant dust condensation efficiency, $f_\mathrm{in}=0.1$, of the ejected metals . [Using the metal production rate per volume, $\dot{\rho}_{\rm Z}$, calculated in the simulations,]{} we write 