We present a novel high-resolution methodology for the numerical solution of compressible flow equations. The resulting schemes are formulated using multidimensional quantities - vorticity and divergence. This is in contrast to the standard methods using the gradients of velocity vector components. The new approach is based upon and introduces the reliance on divergence in addition to vorticity. Two variants of the method are considered: one based upon the Local Lax-Friedrichs flux formula and another one rely upon the HLLC scheme. Besides being more appealing from the conceptual point of view, the new approach will be demonstrated by numerical experiments to be vastly superior in resolving vortical flows and fine structures arising from wave interaction.