Analysis of pipe networks involves computing flow rates and pressure differences on pipe segments in the network, given the external inflow/outflow values. This analysis can be conducted using iterative methods, among which the algorithms of Hardy Cross and Newton-Raphson have historically been applied in practice. In this note, we address the mathematical analysis of the local convergence of these algorithms. The loop-based Newton-Raphson algorithm converges quadratically fast, and we provide estimates for its convergence radius to correct some estimates in the previous literature. In contrast, we show that the convergence of the Hardy Cross algorithm is only linear. This provides theoretical confirmation of experimental observations reported earlier in the literature.

翻译：管道网络分析涉及在给定外部流入/流出量条件下计算网络中管段的流量和压差。此类分析可采用迭代方法进行，其中Hardy Cross算法和Newton-Raphson算法在实践中具有历史应用背景。本研究针对这两种算法的局部收敛性进行数学分析。基于回路的Newton-Raphson算法具有二次收敛速度，我们为其收敛半径提供了估值，以修正先前文献中的部分估计。相比之下，我们证明Hardy Cross算法的收敛性仅为线性。这一结果从理论上印证了文献中早先报道的实验观察结论。