PArAllel cOmPuTATIOnAl AlGOrIThmS In ThermAl PrOceSSeS In meTAllurGY AnD mInInGPurpose. Formation of parallel algorithms in the thermal process simulation in metallurgy and mining. The proposed parallel form of the algorithms must be maximal, and, therefore, have the minimum possible implementation time in parallel computing systems. Elimination of recurrent computation structure of the desired decision vectors, which, as a rule, leads to the rounding errors accumulation. Such class simulation problems are realized by multiprocessor computing systems.methodology. Implementation of parallelizing process of mathematical problem definition is realized by an approach based on the "odd-even" reduction algorithm. The essence of this approach lies in exclusion of simulation coefficients of the process under research by realizing elementary rows transformations of the constructed equations system. A directly parallel form of the algorithm for solving problems is realized by a numerical-analytical approach. It is shown that the compiled parallel form is the maximum, which, in turn, provides minimum time for solving the set problems by multiprocessor computing systems.Findings. The presented research studies in this paper showed a high efficiency of parallelization of systems of tridiagonal structure linear algebraic equations by example of solving thermal problems. The proposed numericalanalytical method for parallelizing tridiagonal systems does not impose any restrictions on the grid nodes topology of the computational domain. With respect to parallel computations of arithmetic expressions, the original data error is separated from the rounding operations by the proposed method. This approach excludes the recurrent structure of the desired decision vectors computation, which, as a rule, leads to accumulation of rounding errors. The proposed parallel form of the algorithms must be maximal, and, therefore, have the minimum possible implementation time in parallel computing systems. Computational experiments conducted by a multiprocessor computer system showed high efficiency of the developed parallel algorithms.Originality. Within decomposition algorithms, based on the "odd-even" reduction method, a new approach to the distributed solution of linear algebraic equation systems is proposed for the first time, which differs from the known methods in the closed parallel form with respect to the central grid node and with a high degree of vectorization. There was proposed, analyzed and implemented a new approach to the solution of metallurgical production problems, which allows increasing economy, productivity and speed of computations. It is proved that this approach provides the highest computation vectorization degree, predetermines the maximum parallel algorithmic form and, as a consequence, the minimum possible time for implementing algorithms on parallel computing systems.Practical value. By using a high-performance multiprocessor system, the developed approach allows processing and interpreting the thermal experiments r...

The work is dedicated to the construction of numerical-analytical method of designing efficient algorithms for the solution of problems in economics and engineering. Using a priori information about the smoothness of the solution, great attention is paid to the construction of high-accuracy solutions. The proposed approach eliminates recurrent structure calculations unknown vectors decisions, which leads to the accumulation of rounding errors. Parallel form of the algorithm is the maximum, and therefore has the shortest possible time the implementation on parallel computing systems. Most conventional algorithms for solving these problems (sweep techniques, decomposition of the matrix into a product of two diagonal matrices, doubling, etc.) when multiple processors work typically no faster than if a single processor. The reason for this is substantial sequence computations of these algorithms.

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Abstract. The article is devoted to distributed simulation of visualization of decision vectors of applied problems on the basis of schemes of increased accuracy order. The higher computational speedup in comparison with the finite-difference approach is illustrated by analytical solutions that allow simultaneous and parallel computations in all temporary layers. It is shown that the most promising approach to mathematical simulation of applied problems is the one that is based on numerical-analytical solutions.Keywords: multiprocessor computing system, speedup, visualization, distributed simulation, numerical-analytical solution.Target setting. Significant computation speedup of applied problems is achieved by means of finitedifference schemes due to the parallelization effect. However, the numerical-analytical algorithms for solving applied problems deserve special attention. Greater computational speedup compared to the finite difference approach can be achieved through analytical solutions that allow simultaneous and parallel computing for all temporary layers and, in this case, do not use combined memory. Thus, the most perspective approach to the mathematical simulation of applied problems should be the one that is based on numerical-analytic solutions. Effective means during the processing of heat and mass transfer tasks in metallurgical industry are considered to be the application of parallel computing technologies on distributed cluster-type systems that have relatively low cost and are easily scaled both by the number of processors and by the amount of RAM [2,12]. Consequently, the distributed simulation of the vector visualization of applied problems solutions on the basis of schemes of the raised accuracy order is an essential and relevant task.Analysis of recent research and publications. Heat and mass transfer processes of metallurgical production should be considered as large systems [8][9][10]. Today, solving complex, large-scale tasks requires powerful computers and is characterized by 'parallel' term, that is, there are parallel computers, computing systems, parallel computing methods, etc. [3 − 5]. In broad terms, this term entered almost immediately after the appearance of the first computers, or rather, after realizing the fact that the computers created were not able to solve, during the optimal term, many practical tasks. The emergence in computing systems of new and expensive communication tools, a more advanced elemental base, stimulated the development of high-performance computations based on multiprocessor computing systems [1, 7].In addition, the class of problems in question is usually solved through set of finite-difference equations, which essence is to replace the derivatives by difference relations. In this case, from the numerical algorithm point of view, the solution of finite-difference equations is divided into explicit and implicit schemes [11]. In an explicit scheme, the values of the desired function are determined sequentially, layer by layer. However, despite the appar...

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