Two finite-difference schemes that preserve the dissipation of energy in a system of modified wave equations

TL;DR

This paper introduces two finite-difference numerical schemes for dissipative sine-Gordon systems, accurately preserving energy dissipation and enabling analysis of nonlinear supratransmission in Josephson junction arrays.

Contribution

The work presents novel conditionally stable finite-difference methods that preserve energy dissipation and accurately approximate energy dynamics in dissipative sine-Gordon systems.

Findings

Schemes effectively estimate the threshold for nonlinear supratransmission.

Methods maintain energy and energy rate approximation accuracy.

Results align with mechanical chain oscillator models.

Abstract

In this work, we present two numerical methods to approximate solutions of systems of dissipative sine-Gordon equations that arise in the study of one-dimensional, semi-infinite arrays of Josephson junctions coupled through superconducting wires. Also, we present schemes for the total energy of such systems in association with the finite-difference schemes used to approximate the solutions. The proposed methods are conditionally stable techniques that yield consistent approximations not only in the domains of the solution and the total energy, but also in the approximation to the rate of change of energy with respect to time. The methods are employed in the estimation of the threshold at which nonlinear supratransmission takes place, in the presence of parameters such as internal and external damping, generalized mass, and generalized Josephson current. Our results are qualitatively in agreement with the corresponding problem in mechanical chains of coupled oscillators, under the presence of the same parameters.