Abstract: The numerical investigation into the instability of theunder-expanded supersonic jets is presented in this paper by solving theaxisymmetric N-S equations with the NND finite difference scheme. Numericalresults demonstrated that there are three kinds of the instability for theunder-expanded supersonic jets according to their shock cell structurepatterns: (1) the regular reflection shock cell with a single shear layer,(2) the Mach reflection shock cell with two shear layers and (3) the curvedMach stem with a relatively high expansion ratio. Moreover, for theunder-expanded supersonic jets, the repetitively-generated quasi-steadyshock cells in the jets indicate their stable development and thedisappearance of these shock cells implies that the jets are about to fallinto instable states.