Analytical Solution of MOC-Based Finite-Difference Water Hammer Equations with Second-Order Implicit Approximation of the Classical Friction Term

Although the classical steady and quasi-steady friction term often overestimates hydraulic head, it remains widely used in water hammer equations to simulate hydraulic transients in piping systems. Its popularity stems from its simple mathematical form and ease of implementation compared to unsteady friction models, particularly in complex networks. Within the method of characteristics (MOC) framework, this classical friction term for turbulent flows is typically integrated using explicit numerical schemes, producing linear finite difference equations that are straightforward and computationally efficient to solve. In contrast, more accurate implicit methods are often avoided due to the perceived need for nonlinear iterative schemes to resolve the resulting equations. This work introduces a unique closed-form analytical solution to the nonlinear algebraic recurrence equation governing discharge in the water hammer problem derived using the Crank–Nicolson approximation of classical turbulent friction. The proposed approach not only delivers second-order accuracy, which is unattainable with conventional explicit approximations, but also remains computationally efficient and robust, offering a superior alternative to existing methods. By evaluating a global normalized absolute error as a function of step size across multiple methods and a representative dimensionless governing parameter, we demonstrate that the improved accuracy offsets the modest increase in central processing unit (CPU) time. Specifically, the proposed methodology consistently reduces global mean errors by up to an order of magnitude compared with other second-order explicit methods, with only approximately a 60% increase in computational cost.