In the left panels ( t = 0 s), the level crossing length at sin $^{2}$2 θ$_{13}$ = 10 − 2 and 10 − 3 are larger than the oscillation length at all region. Therefore, the resonance is adiabatic. On the other hand, in the case of sin $^{2}$2 θ$_{13}$ = 10 − 4 and 10 − $^{5}$, the level crossing length at the resonance is almost same as or slightly smaller than the oscillation length of ¯ ν$_{e}$ . Therefore, the resonance is non-adiabatic. In the right panels ( t = 3 s) when the shock wave reaches the H-resonance, the level crossing length, δr , is of the same order or smaller than the oscillation length L$_{osc}$ for all cases of the mixing angle, expect for sin $^{2}$2 θ$_{13}$ = 10 − $^{2}$. This satisfies non-adiabatic condition γ < 1 as discussed above. We understand, therefore, that the resonance becomes non-adiabatic by the effect of the shock wave. In Figure 2, the influence of the shock wave appears about 2 s after core bounce. This result is consistent with our simulation (see Figure 

1). 

In the case of normal hierarchy, the survival probability of ¯ ν$_{e}$ does not change much because there is no resonance in the ¯ ν sector. Therefore, the survival probability of ¯ ν$_{e}$ in normal hierarchy is always ∼ 0.7 regardless of the value of sin $^{2}$2 θ$_{13}$ or independently of the shock wave. 

We note that there is L-resonance in ν sector even in the case of inverted hierarchy. However, the value of θ$_{12}$ which is related to L-resonance is very large (see Eq. (3)), and the level crossing length of our simulation is not as small as the oscillation length at the resonance. Therefore, γ at the L-resonance is larger than 1. As a result, the survival probability of ν$_{e}$ in inverted hierarchy does not change drastically, and stays always ∼ 0.3. 

## B. Supernova neutrino spectrum 

We calculate the supernova neutrino spectra using the survival probabilities. Figure 4 shows the spectra of ν$_{e}$ . Left panels of Figure 4 show the spectra in the case of normal hierarchy, and right panels show the results of inverted hierarchy. Figure 5 is same as Figure 4 but for ¯ ν$_{e}$ spectra. Red solid and blue dotted lines are the spectra with and without shock wave, respectively. In order to clearly observe the shock wave effects, we display these ratio, φ$_{with}$/φ$_{without}$ , of the spectra with to without shock in lower part of each panel. 

We see clearly the shock wave effects in the ν$_{e}$ spectra in Figure 4 in normal hierarchy. In the case of sin $^{2}$2 θ$_{13}$ = 10 − 3 and 10 − $^{4}$, an enhancement in low energy component of the neutrino spectra is seen when the shock wave reaches the H-resonance. At later times, the 

