  We calculate the charge and spin Drude weight of the one-dimensional extended
Hubbard model with on-site repulsion $U$ and nearest-neighbor repulsion $V$ at
quarter filling using the density-matrix renormalization group method combined
with a variational principle. Our numerical results for the Hubbard model (V=0)
agree with exact results obtained from the Bethe ansatz solution. We obtain the
contour map for both Drude weights in the $UV$-parameter space for repulsive
interactions. We find that the charge Drude weight is discontinuous across the
Kosterlitz-Thouless transition between the Luttinger liquid and the
charge-density-wave insulator, while the spin Drude weight varies smoothly and
remains finite in both phases. Our results can be generally understood using
bosonization and renormalization group results. The finite-size scaling of the
charge Drude weight is well fitted by a polynomial function of the inverse
system size in the metallic region. In the insulating region we find an
exponential decay of the finite-size corrections with the system size and a
universal relation between the charge gap $\Delta_c$ and the correlation length
$\xi$ which controls this exponential decay.
