The High-Performance Parallel Algorithms for the Numerical Solution of Boundary Value Problems

Abstract

The boundary and the initial boundary value problems form the basis of numerous mathematical models. Ultimately, the discrete (linearized) boundary and the initial boundary value problems are reduced to the systems of linear algebraic equations with sparse and ill-conditioned coefficient matrix. In modern applications (such as computational fluid dynamics) the number of equations in the system can reach about $10^{12}_{}$ and higher. Just the numerical solution of such systems requires significant computational effort, so an actual problem of modern computational mathematics is working-out, theoretical analysis and testing of high-performance parallel algorithms. The article discusses algebraic, geometric and combined ways to formation of the parallel algorithms. In this work we presented advantages and disadvantages of each ways, the estimate of parallelism’s acceleration and efficiency, the comparison of volume of computational work compared with the optimal sequential algorithm, and the results of computational experiments. The peculiarities of parallel algorithms’ implementation by using of software and hardware structures for parallel programming were discussed in this work.

The activity is a part of the research work “Supercomputer simulation of physical and chemical processes in the high-speed direct-flow propulsion jet engine of the hypersonic aircraft on solid fuels” supported by Russian Science Foundation (project no. 15-11-30012).