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

  • Vadim Volokhov
  • Sergey MartynenkoEmail author
  • Pavel Toktaliev
  • Leonid Yanovskiy
  • Dmitriy Varlamov
  • Alexander Volokhov
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 753)

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.

Keywords

Initial boundary value problems Parallel algorithms 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Vadim Volokhov
    • 1
  • Sergey Martynenko
    • 1
    • 2
    Email author
  • Pavel Toktaliev
    • 1
    • 2
  • Leonid Yanovskiy
    • 1
    • 2
  • Dmitriy Varlamov
    • 1
  • Alexander Volokhov
    • 1
  1. 1.Institute of Problems of Chemical Physics of RASChernogolovka, Moscow RegionRussian Federation
  2. 2.Central Institute of Aviation Motors n.a. P.I.BaranovMoscowRussian Federation

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