  In this work, we present a numerical method to consistently approximate
solutions of a spatially discrete, double sine-Gordon chain which considers the
presence of external damping. In addition to the finite-difference scheme
employed to approximate the solution of the difference-differential equations
of the model under investigation, our method provides positivity-preserving
schemes to approximate the local and the total energy of the system, in such
way that the discrete rate of change of the total energy with respect to time
provides a consistent approximation of the corresponding continuous rate of
change. Simulations are performed, first of all, to assess the validity of the
computational technique against known qualitative solutions of coupled
sine-Gordon and coupled double sine-Gordon chains. Secondly, the method is used
in the investigation of the phenomenon of nonlinear transmission of energy in
double sine-Gordon systems; the qualitative effects of the damping coefficient
on the occurrence of the nonlinear process of supratransmission are briefly
determined in this work, too.
