Numerical hemodynamic simulations can improve the diagnosis and treatment of cardiovascular-related diseases, decreasing their leading mortality. The paper presents the improvement of the hemodynamic solver, First Blood. While the method of characteristics (MoC) handles the boundary conditions, the MacCormack scheme and the MoC are also applied to solve the mass and momentum balance equations in the inner points. The Olufsen model replaces the Poynting-Thomson model to ensure the accurate modelling of the arterial wall behaviour. The study highlights the numerical superiority of the MacCormack scheme over the MoC both in accuracy and computational efficiency. The simulation of a cardiac cycle takes less than 9 seconds, while the discretisation error in the results is low. Finally, simulation results are presented to validate the physiological relevance of the solver and the applied cardiovascular model.