A pressure-based procedure to solve problems ranging from incompressible to highly compressible flows is described. The method adopts the fully conservative finite-volume approach, for which the meshes can be of any topology. To handle the sharp change of the gradient in the regions near the shock or the solid wall, the convective flux is limited using high-resolution schemes such as the total variation diminishing (TVD) or the normalized variable diagram (NVD) scheme. The flux limiters are determined from the characteristic variables instead of the commonly used primitive or conservative variables. To enhance solution accuracy, the gradient is calculated using a linear reconstruction approach. The method is assessed and validated via testing on a number of flow problems, including inviscid flows through a convergent-divergent nozzle, inviscid and viscous flows past an airfoil, and viscous flows through a double-throat nozzle.