A solution procedure applicable in the range from incompressible to compressible flows is presented. The method is based on the finite-volume approach, for which the computational cells can be polyhedrons of arbitrary geometry. Pressure is employed as a primary dependent variable because of its significant role from very low-speed flows to high-speed supersonic flows. To tackle the abrupt change of gradient in the region of the shock, either the total variation diminishing (TVD) scheme or the normalized variables diagram (NVD) scheme can be incorporated to bound the convective flux. These high-resolution schemes are formulated in the normalized variable formulation (NVF) form and implemented via the form of a TVD flux limiter. The algorithm is tested for inviscid flows at different Mach numbers. The results obtained show that the sharp gradient of the shock can be captured with high resolution. To demonstrate its capability to deal with incompressible flows, calculations are also undertaken for viscous flows over a circular cylinder.