We present a simplified model of data flow on processors in a high-performance computing framework involving computations necessitating inter-processor communications. From this ordinary differential model, we take its asymptotic limit, resulting in a model which treats the computer as a continuum of processors and data flow as an Eulerian fluid governed by a conservation law. We derive a Hamilton–Jacobi equation associated with this conservation law for which the existence and uniqueness of solutions can be proven. We then present the results of numerical experiments for both discrete and continuum models; these show a qualitative agreement between the two and the effect of variations in the computing environment’s processing capabilities on the progress of the modelled computation.

中文翻译：

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并行计算机中异步数据流的数学模型

我们在高性能计算框架中提出了处理器上数据流的简化模型，其中涉及需要进行处理器间通信的计算。从这个普通的微分模型中，我们采用其渐近极限，得到一个模型，该模型将计算机视为处理器的连续体，并将数据流视为受守恒定律控制的欧拉流体。我们推导了与该守恒定律相关的汉密尔顿-雅各比方程，可以证明该方程的存在性和唯一性。然后，我们介绍离散模型和连续模型的数值实验结果；这些显示了两者之间的定性一致性，以及计算环境的处理能力变化对建模计算进度的影响。