  This work is concerned with the construction of Gaussian Beam (GB) solutions
for the numerical approximation of wave equations, semi-discretized in space by
finite difference schemes. GB are high-frequency solutions whose propagation
can be described, both at the continuous and at the semi-discrete levels, by
microlocal tools along the bi-characteristics of the corresponding Hamiltonian.
Their dynamics differ in the continuous and the semi-discrete setting, because
of the high-frequency gap between the Hamiltonians. In particular, numerical
high-frequency solutions can exhibit spurious pathological behaviors, such as
lack of propagation in space, contrary to the classical space-time propagation
properties of continuous waves. This gap between the behavior of continuous and
numerical waves introduces also significant analytical difficulties, since
classical GB constructions cannot be immediately extrapolated to the finite
difference setting, and need to be properly tailored to accurately detect the
propagation properties in discrete media. Our main objective in this paper is
to present a general and rigorous construction of the GB ansatz for finite
difference wave equations, and corroborate this construction through accurate
numerical simulations.
