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Parallel 3D ADI Scheme for Partially Dimension Reduced Heat Conduction Problem
Volume 33, Issue 3 (2022), pp. 477–497
Pub. online: 24 May 2022      Type: Research Article      Open accessOpen Access
Received
1 November 2021
Accepted
1 April 2022
Published
24 May 2022

Abstract

This model describes the heat equation in 3D domains, and this problem is reduced to a hybrid dimension problem, keeping the initial dimension only in some parts and reducing it to one-dimensional equation within the domains in some distance from the base regions. Such mathematical models are typical in industrial installations such as pipelines. Our aim is to add two additional improvements into this methodology. First, the economical ADI type finite volume scheme is constructed to solve the non-classical heat conduction problem. Special interface conditions are defined between 3D and 1D parts. It is proved that the ADI scheme is unconditionally stable. Second, the parallel factorization algorithm is proposed to solve the obtained systems of discrete equations. Due to both modifications the run-time of computations is reduced essentially. Results of computational experiments confirm the theoretical error analysis and scalability estimates of the parallel algorithm.

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Biographies

Čiegis Raimondas
rc@vgtu.lt

R. Čiegis has graduated from Vilnius University Faculty of Mathematics, in 1982. He received the PhD degree from the Institute of Mathematics of Byelorussian Academy of Science in 1985 and the degree of habil. doctor of mathematics from the Institute of Mathematics and Informatics, Vilnius, in 1993. He is a professor and the head of Mathematical Modeling department of Vilnius Gediminas Technical University. His research interests include numerical methods for solving nonlinear PDE, parallel numerical methods, mathematical modelling in nonlinear optics, porous media flows, technology, image processing, biotechnology.

Čiegis Remigijus
remigijus.ciegis@knf.vu.lt

R. Čiegis has graduated from Kaunas Polytechnical Institute Faculty of Chemical Engineering in 1982 and Vilnius University Kaunas Faculty, in 1989. He received the PhD degree in economics from the Vilnius University in 1995 and the degree of habil. doctor of management from the Kaunas Vytautas Magnus University in 2002. He is a professor of Vilnius University. His research interests include environmental economics, sustainable economic development, macroeconomics, regional economics.

Suboč Olga
os@vgtu.lt

O. Suboč has graduated from Vilnius University Faculty of Mathematics, in 1998. She received the PhD degree from the Institute of Mathematics and Informatics and Vytautas Magnus University, in 2002. She is an associated professor in Vilnius Gediminas Technical University. Her research interests include numerical methods for solving nonlinear PDE, mathematical modelling, technology, image processing, non-local conditions, biotechno-logy.


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© 2022 Vilnius University
Open access article under the CC BY license.

Keywords
hybrid dimension model ADI scheme parallel factorization algorithm heat conduction piplines

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