  We apply the Lagrange-mesh method to discretize continuum states of weakly
bound nuclei for continuum-discretized coupled-channel (CDCC) calculations of
three-body breakup reactions. This discretization method is compared with the
bin method, which is regarded as the standard continuum discretization method,
for the $d$ and $^6$Li induced reactions. An improved Numerov algorithm is used
to solve the coupled channels equations, which permits a fast integration of
equations and a convenient treatment of the closed channels. A new CDCC model
code named \emph{CDCC-R} is developed. In all cases, the combination of the
Lagrange-mesh method and the Numerov algorithm shows high efficiency and
accuracy for the CDCC calculations of the elastic scattering and breakup
reactions. Especially, various numerical and physical aspects are discussed for
$^6$Li induced reactions. The coupling effects of the continuum states with
high orbital angular momentum and closed channels are discussed. Moderate
effects are found in the calculations for elastic scattering and breakup
reaction when $F$- and $G$-wave continuum states are included in CDCC model
space for $^6$Li induced reactions at the incident energies well above the
Coulomb barrier. The closed channel effect is found to be indispensable for
$^6$Li breakup reaction calculation when the incident energy is around the
Coulomb barrier.
