  In this work, we present a conditionally stable finite-difference scheme that
consistently approximates the solution of a general class of (3+1)-dimensional
nonlinear equations that generalizes in various ways the quantitative model
governing discrete arrays consisting of coupled harmonic oscillators.
Associated with this method, there exists a discrete scheme of energy that
consistently approximates its continuous counterpart. The method has the
properties that the associated rate of change of the discrete energy
consistently approximates its continuous counterpart, and it approximates both
a fully continuous medium and a spatially discretized system. Conditional
stability of the numerical technique is established, and applications are
provided to the existence of the process of nonlinear supratransmission in
generalized Klein-Gordon systems and the propagation of binary signals in
semi-unbounded, three-dimensional arrays of harmonic oscillators coupled
through springs and perturbed harmonically at the boundaries, where the basic
model is a modified sine-Gordon equation; our results show that a perfect
transmission is achieved via the modulation of the driving amplitude at the
boundary. Additionally, we present an example of a nonlinear system with a
forbidden band-gap which does not present supratransmission, thus establishing
that the existence of a forbidden band-gap in the linear dispersion relation of
a nonlinear system is not a sufficient condition for the system to present
supratransmission.
