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The redistribution and embedding are two main adaptive-grid methods. With use of initial structured grid, it is possible to combine these adaptive-grid techniques. It means that before the grid becomes unstructured, we employ redistribution technique and then use embedding method. It is used to solve three model problems, transonic and supersonic inviscid flows in channels with circular arc bump and with convergent part. The Euler equations are integrated to steady state by an explicit, finite volume, Ni's Lax–Wendroff type. The combination of adaptive-grid redistribution and embedding is complex, but it is shown in this paper that it is very efficient and worth to use it in large CPU time consuming problems. The results also show that the combination requires less grid points and therefore less memory space for prescribed accuracy. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]