A numerical method with properties of consistency in the energy domain for a class of dissipative nonlinear wave equations with applications to a Dirichlet boundary-value problem

摘要

In this work, we present a conditionally stable finite-difference scheme thatconsistently approximates the solution of a general class of (3+1)-dimensionalnonlinear equations that generalizes in various ways the quantitative modelgoverning discrete arrays consisting of coupled harmonic oscillators.Associated with this method, there exists a discrete scheme of energy thatconsistently approximates its continuous counterpart. The method has theproperties that the associated rate of change of the discrete energyconsistently approximates its continuous counterpart, and it approximates botha fully continuous medium and a spatially discretized system. Conditionalstability of the numerical technique is established, and applications areprovided to the existence of the process of nonlinear supratransmission ingeneralized Klein-Gordon systems and the propagation of binary signals insemi-unbounded, three-dimensional arrays of harmonic oscillators coupledthrough springs and perturbed harmonically at the boundaries, where the basicmodel is a modified sine-Gordon equation; our results show that a perfecttransmission is achieved via the modulation of the driving amplitude at theboundary. Additionally, we present an example of a nonlinear system with aforbidden band-gap which does not present supratransmission, thus establishingthat the existence of a forbidden band-gap in the linear dispersion relation ofa nonlinear system is not a sufficient condition for the system to presentsupratransmission.