modified hardy-cross methods with fifth-order convergence

in this study, a new modified fifth-order hardy-cross methods is presented for water distribution systems. in the steady-state condition, the governing equations of water networks are the continuity equations for each node and the energy equations for each loop. the former equations are linear in respect to flow rate while the latter are nonlinear algebraic ones. therefore, the true meaning of solving a pipe network is to solve a system of nonlinear algebraic equations in terms of pipe flow rates. on the contrary, the hardy-cross method, as one of the hydraulic-solver approaches, solves pipe networks without building the aforementioned system of equations, which is supposed to be more efficient than the matrix-based ones. although the original hardy-cross method does not provide a matrix-based scheme, it performs much less efficient in comparison with the other matrix-based approaches. the hardy-cross method utilizes a specific kind of initial guess which satisfies the continuity equations in advance of dealing with the energy equations, which are individually solved for each loop. hence, one of the major challenges of this approach is its low rate of convergence. the proposed modified hardy-cross methods grasp higher order of convergence than the original approach. finally, the comparison of the proposed fifth-order hardy-cross methods with the original one clearly demonstrates that the recommended modification improves the rate of convergence of the original hardy-cross method.