Differential linear repetitive processes are a class of linear systems which can be used, for example, to model industrial processes such as long-wall coal cutting operations. The key feature of interest in this paper is the fact that information propagation in one of the two separate directions in such processes evolves continuously over a finite fixed duration and in the other it is, in effect, discrete. This paper develops discrete approximations for the dynamics of these processes and examines the effects of the approximation techniques used on two key control related properties, i.e. stability and the 2D linear systems structure of the resulting discrete stale space models.