SOLVING A FINITE DIFFERENCE ONE-DIMENSIONAL PROBLEM USING THE PROGONKA METHOD BASED ON AN IMPLICIT SCHEME
Keywords:
finite difference method, implicit difference scheme, one-dimensional problem, heat conduction equation, integro-interpolation method, sweep method, Thomas algorithm, tridiagonal system of equations, boundary conditions, difference grid, non-uniform step, stability, soil temperature, underground structures, mathematical modeling, numerical methods.Abstract
In this article, a method for solving a one-dimensional heat conduction problem using an implicit finite difference scheme based on the integro-interpolation method and the sweep algorithm is considered. A non-uniform spatial difference grid and a uniform temporal grid are constructed, the differential equation is transformed into a discrete form, and as a result, a system of tridiagonal algebraic equations is obtained. Taking boundary conditions into account, an algorithm for determining the solution using the sweep (Thomas) method is presented. The forward and backward sweep stages are consistently analyzed, and the computational stability and efficiency of the method are substantiated. The proposed approach is of practical importance for accurate modeling of the temperature field in soil layers and for assessing the thermal regime of underground structures and foundation constructions.
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