the sign of Δ m 2 13 ≡ m 2 3 − m 2 $_{1}$, and the CP violation phase remain unknown. However, among many studies about implication on these unknown neutrino parameters from future supernova neutrinos [21–29], there are several proposed possibilities that the detection of the neutrinos from the next Galactic supernova would constrain the neutrino oscillation parameters and identify the mass hierarchy more precisely [26–29]. 

Most of the supernova neutrinos are released for about 10 seconds after the core bounce, and interact with electrons when the neutrinos propagate through stellar matter. There- fore, Mikheyev-Smirnov-Wolfstein (MSW) effect is to be taken into account in the neutrino oscillations of the supernova neutrinos. The MSW effect on the supernova neutrino signal has been investigated previously in literatures (e.g., [21, 30–32]). For example, the initial progenitor mass dependence of the early neutrino burst taking account of the MSW effect was studied in [33], and the Earth matter effects of the supernova neutrinos were identi- fied in [34–37]. Moreover, neutrino spin-flavor conversion in supernova has been studied considering numerically in [38–40] and analytically in [41, 42]. 

Recently, the effect of the shock wave on MSW effect of neutrino oscillations in supernova was studied [22, 43]. This effect appears as a decrease of the average energy of ν$_{e}$ in the case of normal mass hierarchy (or ¯ ν$_{e}$ in the case of inverted mass hierarchy) [44, 45]. Since the shock wave passes through H-resonance region, whose typical resonance density is ∼ 10 3 g/cm 3 in a few seconds after core bounce, the adiabaticity of the resonance changes. The time dependence of the neutrino events monitors the density profile of the supernova [26–28, 45, 46]. Therefore, using MSW effects embedded in the supernova neutrinos, we are able to find the density profile of the supernova. However, in many previous studies the influences of the shock wave on the neutrino events were discussed schematically using simplified and parameterized shock-wave profiles [27]. 

There are many supernova simulations taking account of various explosion mechanisms. However, there are a few studies in which the neutrino oscillations were calculated by using simulation results because almost all supernova simulations were performed in only core region although MSW effect occurs at outer envelope of the star. In practice, it is still very hard to carry out multidimensional simulation with neutrino transport to proceed to the shock propagation from the core throughout the envelope. Therefore, it is important for the studies of the neutrino oscillations to calculate the shock propagation throughout a supernova not only in the core but also in the envelope. 

