The converging Richtmyer–Meshkov (RM) instability on single-and dual-mode N2/SF6 interfaces is studied by an upwind conservation element and solution element solver. An unperturbed case is first considered, and it is found that the shocked interface undergoes a long-term deceleration after a period of uniform motion. The evolution of single-mode interface at the early stage exhibits an evident nonlinearity, which can be reasonably predicted by the nonlinear model of Wang et al. (Phys Plasmas 22:082702, 2015). During the deceleration stage, the perturbation amplitude drops quickly and even becomes a negative (phase inversion) before the reshock due to the Rayleigh–Taylor (RT) stabilization. After the reshock, the interface experiences a phase inversion again or does not, depending on the reshock time. The growth of the second-order harmonic in the deceleration stage clearly reveals the competition between the RT effect and the nonlinearity. For dual-mode interfaces, the growth of the first mode (wavenumber k1) relies heavily on the second mode (wavenumber k2) due to the mode coupling effect. Specifically, for cases where k2 is an even or odd multiple of k1, the growth of the first mode is inhibited or promoted depending on its initial amplitude sign and the phase difference between two basic waves, while for cases where k2 is a non-integer multiple of k1, the second mode has negligible influence on the first mode. Through a systematic study, signs of perturbation amplitudes of the generated k2-k1 and k2+k1 waves are obtained for all possible dual-mode configurations, which are reasonably predicted by a modified Haan model (Phys Fluids B 3:2349–2355, 1991).