  The time evolution of electron waves in graphene superlattices is studied
using both microscopic and 'effective medium' formalisms. The numerical
simulations reveal that in a wide range of physical scenarios it is possible to
neglect the granularity of the superlattice and characterize the electron
transport using a simple effective Hamiltonian. It is verified that as general
rule the continuum approximation is rather accurate when the initial state is
less localized than the characteristic spatial period of the superlattice. This
property holds even when the microsocopic electric potential has a strong
spatial modulation or in presence of interfaces between different
superlattices. Detailed examples are given both of the time evolution of
initial electronic states and of the propagation of stationary states in the
context of wave scattering. The theory also confirms that electrons propagating
in tailored graphene superlattices with extreme anisotropy experience virtually
no diffraction.
