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

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.

References in corpus (7)

• Energy transmission in the forbidden bandgap of a nonlinear chain
• Nonlinear Supratransmission
• On the propagation of binary signals in damped mechanical systems of oscillators
• An application of nonlinear supratransmission to the propagation of binary signals in weakly damped, mechanical systems of coupled oscillators
• On the transmission of binary bits in discrete Josephson-junction arrays
• A numerical method for computing radially symmetric solutions of a dissipative nonlinear modified Klein-Gordon equation
• An energy-based computational method in the analysis of the transmission of energy in a chain of coupled oscillators