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

TL;DR

This paper introduces a conditionally stable finite-difference scheme for nonlinear wave equations that accurately approximates energy transfer and demonstrates applications in nonlinear supratransmission and signal propagation in complex systems.

Contribution

It presents a novel energy-consistent numerical method for (3+1)-D nonlinear equations, including stability analysis and practical applications to wave transmission phenomena.

Findings

The scheme accurately approximates energy transfer in nonlinear systems.

Perfect transmission achieved via boundary modulation in generalized Klein-Gordon systems.

Forbidden band-gap systems may not exhibit supratransmission, contrary to previous assumptions.

Abstract

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.