Multiprocessor Computing based Parallel Structures of Mathematical Models of Tridiagonal Systems

The paper is devoted to the construction of parallel forms of mathematical models of a tridiagonal structure. Classical methods for solving mathematical models of a tridiagonal structure with multiprocessor computing systems are processed, as a rule, much slower than those with single processor computing. This circumstance is explained by the recursive approach, which is the basis of classical methods. In this regard, this paper's primary purpose is to construct the most parallel forms when simulating the indicated class of problems. It is convenient to apply the parallelization algorithm and its mapping to parallel computing systems on the two schemes proposed in this paper: finite-difference and numerical analytical. This approach allows arranging separate determination of the thermal physical characteristics of the structures' material, i.e., allows obtaining solutions of coefficient and other inverse problems of thermal conductivity. This paper considers two methods of discretization of differential problems on the example of solving the mathematical physics equation. Moreover, the application of the numerical analytical straight-line method and sweep methods for parallelization of mathematical models with a three-diagonal structure allows constructing its exact node-by-node solutions having the most parallel form and the least implementation time on parallel computing devices. Simultaneously, it is proposed to use finite-difference and numerical-analytical methods in combination with the splitting method as a methodological basis for constructing numerical methods for solving such problems. The splitting method provides an economical and sustainable implementation of numerical models by the scalar sweep method. The proposed approach for the development of methods, algorithms, and software can be applied in various branches of metallurgical thermal physics, economics, as well as environmental problems of the metallurgical industry.