  Extensions of the eikonal approximation to low energy (20MeV/nucleon
typically) are studied. The relation between the dynamical eikonal
approximation (DEA) and the continuum-discretized coupled-channels method with
the eikonal approximation (E-CDCC) is discussed. When Coulomb interaction is
artificially turned off, DEA and E-CDCC are shown to give the same breakup
cross section, within 3% error, of $^{15}$C on $^{208}$Pb at 20MeV/nucleon.
When the Coulomb interaction is included, the difference is appreciable and
none of these models agrees with full CDCC calculations. An empirical
correction significantly reduces this difference. In addition, E-CDCC has a
convergence problem. By including a quantum-mechanical correction to E-CDCC for
lower partial waves between $^{15}$C and $^{208}$Pb, this problem is resolved
and the result perfectly reproduces full CDCC calculations at a lower
computational cost.
