This paper discusses ongoing work on an all-Mach number, non-dissipative algorithm to solve compressible flows on unstructured grids. The algorithm is capable of handling unstructured grids, which allow efficient and speedy generation of computational meshes for complex geometries. The novel feature of the algorithm is an incompressible scaling for pressure that naturally yields the divergence-free condition for velocity as the Mach number goes to zero. The desirable feature of this algorithm is the discrete conservation of kinetic energy in the incompressible flow limit which in turn ensures robustness of the calculation at high Reynolds numbers without the need for any numerical dissipation. We employ a characteristic filter based shock capturing scheme that is applied as a predictor-corrector approach. As a result, shock capturing is active only in the regions of discontinuities, and provides accurate non-dissipative solutions away from the shocks. Results of the algorithm on representative validation flows are presented.