A semi-implicit method for solving the governing equations of axially symmetric, hydrostatic flows in compliant arterial systems is presented. The judicious selection of terms that are discretized implicitly and the use of an Eulerian–Lagrangian method for treating the nonlinear convection terms makes the present formulation stable, relatively simple, and extremely efficient. The resulting method is locally and globally mass conservative. The numerical scheme is able to compute radial velocity profiles for a large range of Reynolds numbers directly from the governing equations of fluid mechanics, without further modelling assumptions. The numerical results obtained for different test problems verify the computational performance and the accuracy of the proposed algorithm. The method has the potential to be used within complex embedded multi-scale models in the future.