  There is a sign that long-duration gamma-ray bursts (GRBs) originate from the
core collapse of massive stars. During a jet puncturing through the progenitor
envelope, high energy neutrinos can be produced by the reverse shock formed at
the jet head. It is suggested that low-luminosity GRBs (LL-GRBs) are possible
candidates of this high energy neutrino precursor up to $\sim {\rm PeV}$.
Before leaving the progenitor, these high energy neutrinos must oscillate from
one flavor to another with matter effect in the envelope. Under the assumption
of a power-law stellar envelope density profile $\rho \propto r^{-\alpha}$ with
an index $\alpha$, we study the properties of ${\rm TeV-PeV}$ neutrino
oscillation. We find that adiabatic conversion is violated for these neutrinos
so we do certain calibration of level crossing effect. The resonance condition
is reached for different energies at different radii. We notice that the
effective mixing angles in matter for ${\rm PeV}$ neutrinos are close to zero
so the transition probabilities from one flavor to another are almost invariant
for ${\rm PeV}$ neutrinos. We plot all the transition probabilities versus
energy of ${\rm TeV-PeV}$ neutrinos from the birth place to the surface of the
progenitor. With an initial flavor ratio
$\phi_{\nu_e}^0:\phi_{\nu_\mu}^0:\phi_{\nu_\tau}^0=1:2:0$, we plot how the
flavor ratio evolves with energy and distance when neutrinos are still in the
envelope, and further get the ratio when they reach the Earth. For ${\rm PeV}$
neutrinos, the ratio is always
$\phi_{\nu_e}:\phi_{\nu_\mu}:\phi_{\nu_\tau}\simeq0.30:0.37:0.33$ on Earth. In
addition, we discuss the dependence of the flavor ratio on energy and $\alpha$
and get a pretty good result. This dependence may provide a promising probe of
the progenitor structure.
