Development of parallel structures of differential tasks of mathematical physics

Abstract

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.