  Loop quantum cosmology homogeneous models with a massless scalar field show
that the big-bang singularity can be replaced by a big quantum bounce. To gain
further insight on the nature of this bounce, we study the semi-discrete loop
quantum gravity Hamiltonian constraint equation from the point of view of
numerical analysis. For illustration purposes, we establish a numerical analogy
between the quantum bounces and reflections in finite difference
discretizations of wave equations triggered by the use of nonuniform grids or,
equivalently, reflections found when solving numerically wave equations with
varying coefficients. We show that the bounce is closely related to the method
for the temporal update of the system and demonstrate that explicit
time-updates in general yield bounces. Finally, we present an example of an
implicit time-update devoid of bounces and show back-in-time, deterministic
evolutions that reach and partially jump over the big-bang singularity.
