In this study, twenty-two new mathematical schemes with third-order of convergence are gathered from the literature and applied to pipe network analysis. The presented methods were classified into one-step, two-step, and three-step schemes based on the number of hypothetical discharges utilized in solving pipe networks. The performances of these new methods and Hardy Cross method were compared by solving a sample pipe network considering four different scenarios (92 cases). The results show that the one-step methods improve the rate of convergence of the Hardy Cross method in 10 out of 24 cases (41%), while this improvement was found to be 39 out of 56 cases (69.64%) and 5 out of 8 cases (62.5%) for the two-step and three-step methods, respectively. This obviously indicates that the modified schemes, particularly the three-step methods, improve the performance of the original loop corrector method by taking lower number of iterations with the compensation of relatively more computational efforts.

中文翻译：

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三阶方案在管网分析中提高Hardy Cross法收敛性的应用

在这项研究中，从文献中收集了22种具有三阶收敛性的新数学方案，并将其应用于管网分析。根据求解管网中假设排放的数量，将提出的方法分为一步，两步和三步方案。通过求解考虑了四种不同情况（92个案例）的样管网络，比较了这些新方法和Hardy Cross方法的性能。结果表明，一步法提高了Hardy Cross方法的收敛速度，其中24例中有10例（41％），而56例中有39例（69.64％）和5例中有改善。两步法和三步法分别为8例（62.5％）。这显然表明修改后的方案，尤其是三步法，