breed of upwind schemes have been produced since the beginning of the 90s. To be practically useful, such a While enjoying demonstrated improvement in accuracy, efficiency, and robustness over existing schemes, the advection up-scheme should hold robustness/stability for a wide range stream splitting method (AUSM) has been found to have deficiencof problems-Euler and Navier-Stokes equations, ideal ies in some cases. This paper describes recent progress toward and nonequilibrium gas, and steady and unsteady flows. improving the AUSM. We show that the improved scheme, termed Our recent attempts toward deriving a scheme meeting AUSM Ο© , features the following properties: (1) exact resolution of these goals have proven to be promising, resulting in two 1D contact and shock discontinuities, (2) positivity preserving of scalar quantity such as the density, (3) free of ''carbuncle phenome-classes of schemes: (1) the so-called AUSM [1, 2] and its non,'' (4) free of oscillations at the slowly moving shock, (5) algorithderivatives, such as AUSMDV [6] by Wada and Liou and mic simplicity, and ( 6) easy entension to treat other hyperbolic the present AUSM Ο© [3], and (2) the HUS [7-9] by Coquel systems. In this paper, we lay out a general construction for the and Liou. The AUSMDV, a blending form of AUSM, flux-AUSM Ο© scheme and prove its heretofore unreported mathematical difference, and flux-vector splittings, improves the roproperties. Especially a CFL-like condition for positivity-preserving bustness of AUSM in dealing with the collision of strong property is derived. This positivity-preserving proves to be tightly related to the capability of calculating strong rarefaction and near shocks. However, the ''carbuncle phenomenon'' appears, vacuum flows. Finally, results of numerical tests on many problems albeit much weaker than that resulting from the Roe are given to confirm the capability and improvements on a variety scheme, and requires a fix. We suggest in [6] an effective of problems including those failed by other well-known procedure to cure the problem. The HUS (hybrid upwind schemes.