Stability of operator-difference schemes with weights for the hyperbolic equation in the space of summable functions with carriers in the network-like domain

Abstract

This work is a natural continuation of the authors’ research on flow phenomena in the direction of increasing the dimensionality of the network-like domain of change of the space variable. The possibility of practical use of the analysis of the stability of operator-difference schemes to solve the issue of stability (stabilization) of wave phenomena in the engineering of the process of transferring continuous media through network-like carriers (water pipelines, gas and oil pipelines, industrial carriers of petroleum products) is shown. Namely, if the scheme is stable, then sufficiently small changes in the initial data of the mathematical model of the process imply small changes of the solution of the difference problem, i. e. in practice do not lead to undesirable aftereffects. If the schema is unstable, then small changes to the initial data can lead to arbitrarily large changes of the solution. In the process of exploitation of industrial constructions of network-like carriers, wave phenomena inevitably arise, the consequence of which are various kinds of instabilities that entail destruction of one nature or another. It is possible to avoid or essentially reduce such undesirable oscillations using the analysis of the stability properties of the mathematical model of the wave process. The obtained results are used in the algorithmically and digitalization of modern technological processes of the movement of fluid media and gases.