Choosing the optimal multi-point iterative method for the Colebrook flow friction equation -- Numerical validation

TL;DR

This paper evaluates various multi-point iterative methods to efficiently solve the implicit Colebrook flow friction equation, identifying those that require the fewest iterations for accurate solutions in engineering applications.

Contribution

It systematically compares advanced iterative methods and recommends the most efficient ones for solving the Colebrook equation with minimal iterations.

Findings

Most powerful three-point methods need only two iterations.

Recommended methods include Sharma-Guha-Gupta and others based on Kung-Traub and Steffensen schemes.

The approach balances iterative procedures with explicit approximations for engineering use.

Abstract

The Colebrook equation $\zeta$ is implicitly given in respect to the unknown flow friction factor $\lambda$; $\lambda=\zeta(Re,\epsilon^*,\lambda)$ which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. Common approach to solve it is through the Newton-Raphson iterative procedure or through the fixed-point iterative procedure. Both requires in some case even eight iterations. On the other hand numerous more powerful iterative methods such as three-or two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require in worst case only two iterations to reach final solution. The recommended representatives are Sharma-Guha-Gupta, Sharma-Sharma, Sharma-Arora, D\v{z}uni\'c-Petkovi\'c-Petkovi\'c; Bi-Ren-Wu, Chun-Neta based on Kung-Traub, Neta, and Jain method based on Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations.