We present a class of simple advection-pressure splitting numerical methods to solve the blood flow equations in compliant arterial vessels. The schemes are inspired by the TV flux vector splitting approach for conservative systems, proposed by Toro and Vázquez [30]. But the reformulated TV-type splitting schemes of this paper have a wider range of applicability, including systems of equations in non-conservative form. The spatial differential operator is split into advection terms, which may be in conservative form, from pressure terms in conservative or non-conservative form. Additionally, unlike the original TV scheme, the reformulated splitting of this paper fully preserves the continuity equation as part of the pressure system. This last feature is consistent with zero-dimensional models for blood flow that are based on neglecting the inertial term in the momentum equation. The schemes are also well suited for systems in which geometric and biomechanical parameters of the problem vary discontinuously. The splitting schemes of this paper are systematically assessed on a carefully designed suite of test problems and compared with several existing, mainstream methods. Overall, the proposed numerical methods perform very satisfactorily and suggest themselves as attractive computational tools for modelling the dynamics of bodily fluids under realistic conditions.