  We investigate relativistic flows after a shock wave generated in a star
arrives at the surface. First, the sphericity effect is involved through a
successive approximation procedure by adding correction terms to an already
known self-similar solution in the ultra-relativistic limit assuming the plane
parallel geometry. We also find that the involvement of sphericity increases
the acceleration in the early phase as compared with that of the original
plane-parallel flow. Second, we obtain semi-analytic solutions for a mildly
relativistic flow in which the rest mass energy density is not negligible in
the equation of state. To take into account this, we use the enthalpy and the
pressure instead of using the density and the pressure as thermodynamic
variables. These solutions assume self-similar evolutions except for the
initial conditions. Third, we have carried out numerical calculations with a
special-relativistic hydrodynamical code based on the Godunov method in order
to check the applicability of the above sphericity corrections and the
semi-analytic solutions. The equation of state used in our calculations
includes the rest mass energy density. Comparisons with results of numerical
calculations support the validity of the sphericity correction terms. The
evolutions of the pressure and the Lorentz factor of each fluid element of the
semi-analytic solution for mildly relativistic flows match the numerical
results at least in early phases. We also investigate the final free expansion
phases by this code. We have derived the final energy distributions for these
flows and compare them with previous works.
