the change of the grain size distribution by stellar dust production as $$\begin{aligned}
\left[\frac{\upartial\rho_\mathrm{d}(m,\, t)}{\upartial t}\right]_\mathrm{star}=
f_\mathrm{in}\dot{\rho}_Z\, m\tilde{\varphi} (m),\label{eq:stellar}\end{aligned}$$ where $m\tilde{\varphi} (m)$ is the mass distribution function of the dust grains produced by stars, and it is normalized so that the integration for the whole grain mass range is unity. The grain size distribution of the dust grains produced by stars \[which is related to the above mass distribution as $\varphi (a)\,\mathrm{d}a\equiv\tilde{\varphi}(m)\,\mathrm{d}m$\] is written as $$\begin{aligned}
{\varphi}(a)=\frac{C_\varphi}{a}\exp\left\{
-\frac{[\ln (a/a_0)]^2}{2\sigma^2}\right\} ,\end{aligned}$$ where $C_\varphi$ is the normalization factor, $\sigma$ is the standard deviation, and $a_0$ is the central grain radius. We adopt $\sigma =0.47$ and $a_0=0.1\,\micron$ following [@2013MNRAS.432..637A].

### Dust destruction and growth {#subsubsec:destruction}

As shown by HA19, dust destruction by sputtering in SN shocks and dust growth by accretion are both described by the following same form of equation: $$\begin{aligned}
\left[\frac{\upartial\rho_\mathrm{d}(m,\, t)}{\upartial t}\right]_\mathrm{sput/acc}
=-\frac{\upartial}{\upartial m}[\dot{m}\rho_\mathrm{d}(m,\, t)]+
\frac{\dot{m}}{m}\rho_\mathrm{d}(m,\, t).\label{eq:sputtering}\end{aligned}$$ where $\dot{m}\equiv\mathrm{d}m/\mathrm{d}t$. We estimate that $$\begin{aligned}
\dot{m}=\xi (t)m/\tau_\mathrm{dest/acc}(m),\end{aligned}$$ where $\tau_\mathrm{dest/acc}(m)$ is the grain-mass-dependent time-scale of destruction or accretion. In the actual computations, we solve the effects of destruction and accretion, separately. For destruction, we formally choose $\xi (t)=-1$. For accretion, $\xi (t)$ is the fraction of metals in the gas-phase \[i.e. the fraction 