  We calculate the self-force acting on a particle with scalar charge moving on
a generic geodesic around a Schwarzschild black hole. This calculation requires
an accurate computation of the retarded scalar field produced by the moving
charge; this is done numerically with the help of a fourth-order convergent
finite-difference scheme formulated in the time domain. The calculation also
requires a regularization procedure, because the retarded field is singular on
the particle's world line; this is handled mode-by-mode via the mode-sum
regularization scheme first introduced by Barack and Ori. This paper presents
the numerical method, various numerical tests, and a sample of results for
mildly eccentric orbits as well as ``zoom-whirl'' orbits.
