our previous work [@takiwaki12] in which the spectral neutrino transport scheme IDSA [@idsa] was implemented in the ZEUS-MP code [@hayes].

In the original IDSA scheme, a steady-state approximation ($\partial {f^{\rm s}}/(\partial t) = 0$) is assumed. Here $f^{\rm s}$ represents the [*streaming*]{} part of the neutrino distribution function (e.g., @idsa). Then one should deal with a Poisson-type equation to find the solution of $f^s$ (e.g., Eq.(10) in @idsa). This is computationally expensive, because a collective data-communication is required on the MPI routines for all the processors (along the given radial direction in the ray-by-ray approximation) to solve the Cauchy problem.

To get around the problem, one needs to solve the evolution of $f^{\rm s}$ as, $$\begin{aligned}
\frac{\partial \mathcal{E}^{\rm s}}{c\partial {t}} &+&
\frac{1}{r^2}\frac{\partial}{\partial r}r^2 \mathcal{F}^{\rm s}
= \mathcal{S}[{j}, {\chi}, {\Sigma}],\label{eq:fs-evol}\\
\mathcal{E}^{\rm s} &\equiv& \frac{1}{2}\int d{\mu}\,{f}^{{\rm s}},\\
\mathcal{F}^{\rm s} &\equiv& \frac{1}{2}\int \mu d{\mu}\,{f}^{{\rm s}},\\
\mathcal{S} &\equiv& -\left( \hat{j}+\hat{\chi}\right)\mathcal{E}^{\rm s}+\Sigma,\end{aligned}$$ where $\mathcal{E}^{\rm s}$ and $\mathcal{F}^{\rm s}$ corresponds to the radiation energy and flux of the streaming particle, and $\mathcal
S$ represents the source term that is a functional of the neutrino emissivity ($j$), absorptivity ($\chi$), and the isotropic diffusion term ($\Sigma$) all defined in the laboratory frame, respectively. From local hydrodynamic quantities (density, 