Локально-одномерный метод для уравнения переноса сплошной среды с распределенными параметрами на сетеподобной области
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Научный журнал Моделирование, оптимизация и информационные технологииThe scientific journal Modeling, Optimization and Information Technology
Online media
issn 2310-6018

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

Tran D.  

UDC 517.977.56
DOI: 10.26102/2310-6018/2022.37.2.008

  • Abstract
  • List of references
  • About authors

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.

1. Tran D., Provotorov V.V. Finite difference method for transfer equation with distributed parameters on the network. Modelirovaniye, optimizatsiya i informatsionnyye tekhnologii = Modeling, Optimization and Information Technology. 2021;9(3). Available from: https://moitvivt.ru/ru/journal/pdf?id=1019 (accessed on 05/02/2022). DOI: 10.26102/2310-6018/2021.34.3.012 (In Russ.)

2. Tikhonov A.N., Samarskii A.A. Uravneniya matematicheskoi fiziki. Izd. 5-e. M. Nauka; 1977. 736 p. (In Russ.)

3. Kalitkin N.N Chislennye metody. Glavnaja redakcija fiziko-matematicheskoj literatury iz-va «Nauka». M.; 1978. (In Russ.)

4. Provotorov V.V., Provotorova E.N. Optimal control of the linearized Navier-Stokes system in a netlike domain. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes. 2017;13(4):431–443. Available at: https://doi.org/10.21638/11702/spbu10.2017.409 (accessed on 05/02/2022).

5. Artemov M.A., Baranovskii E.S., Zhabko A.P., Provotorov V.V. On a 3D model of non-isothermal flows in a pipeline network. Journal of Physics. Conference Series. 2019;(1203). Article ID 012094. Available at: https://doi.org/10.1088/1742-6596/1203/1/012094 (accessed on 05/02/2022).

6. Zhabko A.P., Provotorov V.V., Balaban O.R. Stabilization of weak solutions of parabolic systems with distributed parameters on the graph. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes. 2019;15(2):187–198. Available at: https://doi.org/10.21638/11702/spbu10.2019.203 (accessed on 05/02/2022).

7. Zhabko A.P., Nurtazina K.B., Provotorov V.V. About one approach to solving the inverse problem for parabolic equation. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes. 2019;15(3):323–336. Available at: https://doi.org/10.21638/11702/spbu10.2019.303 (accessed on 05/02/2022).

8. Tran D., Part A.A. Parametric optimization of the continuous medium transferring process over a network carrier. Modelirovaniye, optimizatsiya i informatsionnyye tekhnologii = Modeling, Optimization and Information Technology. 2021;9(4). Available from: https://moitvivt.ru/ru/journal/pdf?id=1090 (accessed on 05/02/2022). DOI: 10.26102/2310-6018/2021.35.4.037. (In Russ.)

9. Sergeev S.M., Sidnenko T.I., Sidnenko D.B. Distribution centers for agriculture, their modeling. International Scientific School «Paradigma» Summer-2016 Selected Papers. Yelm, WA, USA. 2016;92–97 (accessed on 05.02.2022).

10. Iliashenko O., Sergeev S., Krasnov S. Calculation of high-rise construction limitations for non-resident housing fund in megacities. E3S Web of Conferences. 2018;03006 (accessed on 05/02/2022).

Tran Duy

Email: tranduysp94@gmail.com

Voronezh State University, Voronezh, Russian Federation

Voronezh, Russia

Keywords: initial-boundary value transfer problem, network (directed graph), continuous medium transfer, difference scheme, locally one-dimensional method

For citation: Tran D. Locally one-dimensional method for the transfer equation of a continuous medium with distributed parameters on a network-like domain. Modeling, Optimization and Information Technology. 2022;10(2). Available from: https://moitvivt.ru/ru/journal/pdf?id=1141 DOI: 10.26102/2310-6018/2022.37.2.008 (In Russ).

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Full text in PDF

Received 15.03.2022

Revised 19.04.2022

Accepted 28.04.2022

Published 11.05.2022