Locally one-dimensional method for the transfer equation of a continuous medium with distributed parameters on a network-like domain

The paper considers a wide range of issues related to the solution of an initial-boundary value problem for a parabolic partial differential equation with a multidimensional space variable belonging to the Euclidean space and changing on a network-like domain. The mathematical model describing the process of transferring a continuous medium over a network carrier is determined by the formalism of the initial-boundary value problem. An idea that has become classical is further developed for the case when a network-like region is a directed bounded graph, i.e., a collection of a finite number of segments connected to each other by means of end points. The study employs classical approximations of evolutionary differential equations of the 2-nd order as well as non-classical approximations of differential relations illustrated by generalized Kirchhoff conditions at the branching points of a network-like region (nodal points of the region). When using difference approximations of the initial-boundary value problem operator, the approximation error and stability conditions for the difference scheme are established. The characteristic properties of the locally one-dimensional method and the sweep method utilized to solve the stated problem are studied. An algorithm for the numerical solution of the stated problem is proposed, a computer program is designed, and a computational experiment is carried out on a series of applied problems. The findings are of interest in the analysis of applied problems of multiphase continuum media transfer along network-like 3D carriers.

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