Wave energy converters’ mooring systems are subjected to highly dynamic motions responsible for complex loading mechanisms. Snap loads differ from other dynamic load types by the propagation of tension discontinuities along the cable, which can lead to premature mooring failures if not correctly predicted. Accurate prediction of these loads requires numerical models that can resolve non-smooth solutions. This work developed a (conservative) finite-volume formulation for solving the hyperbolic cable dynamics equation by applying the high-order non-oscillatory central-upwind scheme to calculate the numerical fluxes at the cell interfaces. The central-upwind scheme is a variant of the Godunov-type central finite-volume method and does not need a Riemann solver. This new numerical formulation was tested against analytical and experimental data for a single catenary model under snap-loading conditions. It is shown the method’s ability to simulate snap loads correctly, predicting motions and tensions. Furthermore, it exhibits second-order accuracy in space for smooth solutions. This approach is highly robust (with a numerical dissipation independent of the time step) and simpler to implement compared to other codes in the literature.