An implicit difference scheme for the long-time evolution of localized solutions of a generalized Boussinesq system
type: Item
instance of: scholarly article
title: An implicit difference scheme for the long-time evolution of localized solutions of a generalized Boussinesq system
zbMATH Open document ID: 0822.65118
published in: Journal of Computational Physics
publication date: 1995-04-05T00:00:00Z
review text: For the longitudinal displacement and the transversal one in the one- dimensional model of an atomic chain the authors obtain a system of differential-difference equations. As the continuum limit of this system they get a generalized Boussinesq-like equation coupled to a linear wave equation. Under some auxiliary assumptions and appropriate boundary conditions both discrete and continuous problems are treated numerically by a fully implicit difference scheme employing Newton's quasilinearization. The scheme is stable for a wide range of time steps and initial conditions and as a seven-diagonal it turns out to be well prepared for solving. Next, a decoupled case, i.e. the sole Boussinesq-like equation is investigated numerically for various initial conditions and for different values of time increment. Two distinct classes of solutions appear then. The first one consists of oscillatory pulses whose envelopes are localized waves. The second one consists of solutions whose shapes are either heteroclinic or homoclinic. They exhibit many interesting features. After a suitable rescaling of the pure Boussinesq-like equation the authors numerically confirm the existence of self-similar (in some sense) solutions. Finally, the coupled system is carefully considered by calculations for many combinations of initial conditions and parameters describing the system. Results of the paper seem to be important for metallurgists.
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35R10
zbMATH DE Number: 740608
zbMATH Keywords: long-time evolution
zbMATH Keywords: stability
zbMATH Keywords: localized solutions
zbMATH Keywords: differential- difference equations
zbMATH Keywords: Boussinesq-like equation
zbMATH Keywords: linear wave equation
zbMATH Keywords: fully implicit difference scheme
zbMATH Keywords: Newton's quasilinearization
author: C. I. Christov
author: Gérard A. Maugin
reviewed by: S. Burys
MaRDI profile type: Publication
full work available at URL: jcph.1995.1004
OpenAlex ID: W2084396382
DOI: 10.1006/JCPH.1995.1004
description: scientific article; zbMATH DE number 740608