  A new method for solving the time-dependent two-center Dirac equation is
developed. The time-dependent Dirac wave function is represented as a sum of
atomic-like Dirac-Sturm orbitals, localized at the ions. The atomic orbitals
are obtained by solving numerically the finite-difference one-center Dirac and
Dirac-Sturm equations with the potential which is the sum of the exact
reference-nucleus potential and a monopole-approximation potential from the
other nucleus. An original procedure to calculate the two-center integrals with
these orbitals is proposed. The approach is tested by calculations of the
charge transfer and ionization cross sections for the H(1s)--proton collisions
at proton energies from 1 keV to 100 keV. The obtained results are compared
with related experimental and other theoretical data. To investigate the role
of the relativistic effects, the charge transfer cross sections for the
Ne^{9+}(1s)--Ne^{10+} (at energies from 0.1 to 10 MeV/u) and
U^{91+}(1s)--U^{92+} (at energies from 6 to 10 MeV/u) collisions are calculated
in both relativistic and nonrelativistic cases.
