An Efficient Algorithm for the Numerical Solution of a Three-dimensional Thermal Conductivity Issue
( Pp. 56-62)

A mathematical model of slab heating based on a numerical solution of a three-dimensional thermal conductivity problem has been created and programmatically implemented in the work. To solve a system of difference equations, a layer-by-layer method is proposed that allows using the run-through method to solve a system of difference equations of a three-dimensional problem within a rapidly converging iterative procedure. Comparative calculations of slab heating are carried out using the proposed layer-by-layer method and the method of simple iteration with different degrees of grinding of the spatial grid. As a result, it was found that with the grinding of the mesh, the effectiveness of the layered method in relation to the simple iteration method increases.

Vargin A.V., Levitskiy I.A. An Efficient Algorithm for the Numerical Solution of a Three-dimensional Thermal Conductivity Issue. Computational Nanotechnology. 2023. Vol. 10. No. 4. Pp. 56–62. (In Rus.) DOI: 10.33693/2313-223X-2023-10-4-56-62. EDN: HNVSWA

Timoshpolskiy V.I., Trusova I.A., Mendeleev D.V., Ratnikov P.E. Analysis of methods of mathematical modeling of heat transfer processes in industrial furnaces for metal heating. Casting and metallurgy. 2012. Vol. 2. No. 65. Pp. 102–107. (In Rus.).
Kurnosov V.V., Levitskii I.A., Pribytkov I.A. Massive billets different rates heating in batch furnaces study. Izvestiya. Ferrous Metallurgy. 2012. Vol. 55. No. 9. Pp. 27–31. (In Rus.)
Ganul A.O., Dozhdikov V.I., Mordovkin D.S. The mathematical model of heating the metal in the continuous furnace taking into account the preliminary air heating. Izvestiya. Ferrous Metallurgy. Bulletin of Scientific, Technical and Economic Information. 2018. No. 1 (1). Pp. 102–105. (In Rus.)
Arutyunov V.A., Bukhmirov V.V., Krupennikov S.A. Mathe­matical modeling of thermal operation of industrial furnaces. Moscow: Metallurgiya. 1990. 239 p.
Tikhonov A.N., Samarskii A.A. Equations of mathematical physics. Moscow: Nauka. 2014. 724 p.
Samarskii A.A. Introduction to numerical methods. Moscow: Nauka. 1997. 239 p.
Dulnev G.N., Parfenov V.G., Sigalov A.V. The use of com­puters for solving heat transfer problems. Moscow: Higher School. 1990. 207 p.
Abdukodirov I.B., Vargin A.V., Levitskii I.A. Mathematical model of slab heating in a furnace with walking beams. News of Higer Educations School. Ferrous Metallurgy. Bulletin of Scientific, Technical and Economic Information. 2023. Vol. 66. No. 1. Pp. 112–118. (In Rus.)

mathematical modeling, three-dimensional thermal conductivity issue, finite difference method, tridiagonal coefficient matrix, run-through method.