Choosing the optimal multi-point iterative method for the Colebrook flow friction equation -- Numerical validation
For engineers needing fast and accurate solutions to the Colebrook equation, this work provides a practical selection of optimal iterative methods, though the improvement is incremental over existing numerical techniques.
The authors identify multi-point iterative methods that solve the Colebrook equation for flow friction in as few as two iterations, outperforming Newton-Raphson and fixed-point methods which require up to eight iterations. Recommended methods include Sharma-Guha-Gupta, Sharma-Sharma, and others.
The Colebrook equation $ζ$ is implicitly given in respect to the unknown flow friction factor $λ$; $λ=ζ(Re,ε^*,λ)$ 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, Džunić-Petković-Petković; 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.