  The low-dissipation central-upwind (LDCU) schemes have been recently
introduced in [A. Kurganov and R. Xin, J. Sci. Comput., 96 (2023), Paper No.
56] as a modification of the central-upwind (CU) schemes from [{\sc A. Kurganov
and C. T. Lin, Commun. Comput. Phys., 2 (2007), pp. 141-163}]. The LDCU schemes
achieve much higher resolution of contact waves and many (two-dimensional)
structures resulting from complicated wave interaction. However, the LDCU
schemes sometimes produce more oscillatory results compared with the CU
schemes, especially near the computational domain boundaries.
  In this paper, we propose a very simple -- yet systematic -- modification of
the LDCU schemes, which completely eliminates the aforementioned oscillations
almost without affecting the quality of the computed solution.
