Application of Third-Order Schemes to Improve the Convergence of the Hardy Cross Method in Pipe Network Analysis

Table 1

Summary of the third-order convergence algorithms.

Algorithms	Year	No. of steps	No. of function evaluation	No. of first-derivative evaluation	No. of second-derivative evaluation
(a) One-step modified schemes
Chebyshev’s algorithm	1964	1	1	1	1
Halley’s algorithm	1995	1	1	1	1
Abbasbandy’s algorithm	2003	1	1	1	0
Chun’s first algorithm	2006	1	1	1	0
Chun and Kim’s first algorithm	2010	1	1	1	1
Chun and Kim’s third algorithm	2010	1	1	1	1
(b) Two-step modified schemes
Weerakoon and Fernando’s algorithm	2000	2	1	2	0
Frontini and Sormani’s third algorithm	2003	2	1	2	0
Amat et al.’s algorithm	2003	2	2	1	0
Özban’s algorithm	2004	2	1	2	0
Kou et al.’s algorithm	2006	2	2	1	0
Chun’s second algorithm	2006	2	2	2	0
Chun’s third algorithm	2006	2	2	1	0
Jisheng et al.’s first algorithm	2006	2	1	2	0
Jisheng et al.’ second algorithm	2006	2	1	2	0
Darvishi and Barati’s algorithm	2007	2	2	1	0
Zhou’s algorithm	2007	2	1	1	0
Ham et al.’s algorithm	2008	2	2	1	0
Chun and Kim’s second algorithm	2010	2	1	2	0
Chun and Kim’s fourth algorithm	2010	2	1	2	0
(c) Three-step modified schemes
Jisheng et al.’s third algorithm	2006	2	1	2	0
Zavalani’s algorithm	2014	2	1	2	0