Unification of approaches to the numerical solution of the boundary value problem for heat conduction using TDMA

Modelling heat and mass transfer processes is essential in designing and optimising technological processes in power engineering, mechanical engineering, metallurgy, chemical industry, and other engineering fields. For the mathematical description of such processes, differential equations of heat conduction and diffusion are used, the solution of which requires the application of efficient numerical methods, especially in the case of complex geometries and diverse boundary conditions. This study presents a unified methodology for the numerical solution of boundary value problems of heat conduction with internal heat sources, based on locally one-dimensional implicit finite difference schemes derived using the integral-interpolation method (balance method) in Cartesian and cylindrical coordinate systems. Special attention is given to discretising boundary conditions of the first, second, and third kinds, focusing on Robin conditions, the most commonly encountered in engineering practice. A quasi-linear approximation scheme and spatial splitting schemes are recommended to increase the efficiency of numerical solutions. This approach enables the application of the unconditionally stable Tridiagonal Matrix Algorithm (TDMA). The introduction of indica-tor coefficients provides flexibility in implementation, allowing the balance equation to be used variably by manipulating the terms responsible for heat fluxes and the location of com-putational nodes. This ensures ease of implementation and improves code readability, facili-tating software development for computational modelling. The results of the numerical simu-lation obtained using the proposed method are compared with known analytical and numerical solutions and demonstrate high accuracy. The proposed methodology opens broader opportu-nities for modelling thermal regimes in complex engineering systems.