  In [Cou15] a multiplier technique, going back to Leray and G{\aa}rding for
scalar hyperbolic partial differential equations, has been extended to the
context of finite difference schemes for evolutionary problems. The key point
of the analysis in [Cou15] was to obtain a discrete energy-dissipation balance
law when the initial difference operator is multiplied by a suitable quantity.
The construction of the energy and dissipation functionals was achieved in
[Cou15] under the assumption that all modes were separated. We relax this
assumption here and construct, for the same multiplier as in [Cou15], the
energy and dissipation functionals when some modes cross. Semigroup estimates
for fully discrete hy-perbolic initial boundary value problems are deduced in
this broader context by following the arguments of [Cou15].
