  The modified Hunter--Saxton equation models the propagation of short
capillary-gravity waves. As it involves a mixed derivative, its initial value
problem on the periodic domain is much more complicated than the standard
evolutionary equations. Although its local well-posedness has recently been
proved, the behavior of its solution is yet to be investigated. In this paper,
to develop a reliable numerical method for this problem, we derive a
conservative finite difference scheme. Then, we rigorously prove not only its
stability in the sense of the uniform norm but also its uniform convergence to
sufficiently smooth exact solutions. Discrete conservation laws are used to
overcome the difficulty due to the mixed derivative.
