Newton-Raphson methods
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Another family of powerful and widely used iterative techniques for solving the systems of non-linear equations encountered in network hydraulics is the Newton-Raphson method (and its variants). If you've encountered this method in other areas of engineering or mathematics for finding roots of equations, you'll recognize the underlying principle. This section will explain how this principle is extended to solve the multiple, interconnected equations of a pipe network.
The core idea is to use a Taylor series expansion to linearize the non-linear network equations around an estimated solution point at each iteration. This converts a difficult non-linear problem into a more manageable (though still large) system of linear equations that can be solved using standard matrix techniques.
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