This note gives a numerical investigation for the popular high resolution conservativeschemes when applied to inviscid, compressible, perfect gas °ows with an initial highdensity ratio as well as a high pressure ratio. The results show that they work veryine±ciently and may give inaccurate numerical results even over a very ?ne meshwhen applied to such a problem. Numerical tests show that increasing the order ofaccuracy of the numerical schemes does not help much in improving the numericalresults. How to cure this di±culty is still open.