EFFICIENT ALGORITHMS FOR PARALLELIZING TRIDIAGONAL SYSTEMS  OF EQUATIONS

The article is devoted to the development of the maximal parallel forms of mathematical models with a tridiagonal structure. The example of solving the Dirichlet and Neumann problems by the method of straight lines and the sweep method for the heat equation illustrates the direct fundamental features of constructing parallel algorithms. It is noted that the study of the heat and mass transfer processes is run through their numerical modeling based on modern computer technology.It is shown that with the multiprocessor computing systems’ development, there disappear the problems of increasing their peak performance. On the other hand, building such systems, as a rule, requires standard network technologies, mass-produced processors, and free software. The noted circumstances aim at solving the so-called big problems.It should be borne in mind that the classical approach to solving the tridiagonal structure models based on multiprocessor computing systems is far more time-consuming compared to single-processor computing facilities. That is explained by the recurrence relations that make the basis of classical methods. Therefore, the proposed studies are relevant and aim at the distributed algorithms development for solving applied problems.The proposed research aims to construct the maximal parallel forms of mathematical models with a tridiagonal structure.The paper proposes the schemes to implement parallelization algorithms for applied problems and their mapping to parallel computing systems.Parallelization of tridiagonal mathematical models by the method of straight lines and the sweeping method allows designing absolutely stable algorithms with the maximum parallel form and, therefore, the minimum possible time for their implementation on parallel computing devices. It is noteworthy that in the proposed algorithms, the computational errors of the input data are separated from the round-off errors inherent in a PC.The proposed approach can be used in various branches of metallurgical, thermal physics, economics, and ecology problems in the metallurgical industry.

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Development of parallel structures of differential tasks of mathematical physics

System technologies ◽

10.34185/1562-9945-3-128-2020-04 ◽

2020 ◽

Vol 3 (128) ◽

pp. 36-45

Author(s):

Gennady Shvachych ◽

Volodymyr Konovalenkov ◽

Olena Ivaschenko ◽

Larysa Sushko

Keyword(s):

Parallel Computing ◽

Finite Difference ◽

Mathematical Models ◽

Numerical Models ◽

Splitting Method ◽

Metallurgical Industry ◽

Mathematical Physics ◽

Separate Determination ◽

Sequential Method ◽

Sustainable Implementation

The paper is devoted to the construction of parallel forms of mathematical models of a tridiagonal structure. Two methods of discretization of differential problems are considered by the example of solving the mathematical physics equation. Moreover, the application of the numerical-analytical straight line method and sweep methods to parallelization of mathematical models with a tridiagonal structure allows constructing its exact node-by-node solutions with the maximum parallel form and the least implementation time on parallel computing devices. This paper proposes to apply 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. For such systems, acceptable acceleration in most cases is achieved by parallelizing operations in the corresponding sequential method, forming linear sections.It is convenient to implement 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 thermophysical characteristics of the structures’ material, i.e. allows obtaining solutions of coefficient and other inverse problems of thermal conductivity.The proposed approach to the development of methods, algorithms and software can be applied in various branches of metallurgical thermal physics, economics, as well as for environmental problems of the metallurgical industry.

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