  In this report it is shown that the implicit Euler time-discretization of
some classes of switching systems with sliding modes, yields a very good
stabilization of the trajectory and of its derivative on the sliding surface.
Therefore the spurious oscillations which are pointed out elsewhere when an
explicit method is used, are avoided. Moreover the method (an {\em
event-capturing}, or {\em time-stepping} algorithm) allows for accumulation of
events (Zeno phenomena) and for multiple switching surfaces (i.e., a sliding
surface of codimension $\geq 2$). The details of the implementation are given,
and numerical examples illustrate the developments. This method may be an
alternative method for chattering suppression, keeping the intrinsic
discontinuous nature of the dynamics on the sliding surfaces. Links with
discrete-time sliding mode controllers are studied.
