LAUSR

In this work, the single-mode Richtmyer–Meshkov instability is studied numerically to find a reasonable nonlinear theoretical model which can be applied to predict the interface evolution from the linear stage to the early nonlinear stage. The cut-cell-based sharp-interface methods $$\hbox {MuSiC}^{+}$$MuSiC+ (Chang et al. in J Comput Phys 242:946–990, 2013) and CCGF (Bai and Deng in Adv Appl Math Mech 9(5):1052–1075, 2017) are applied to generate numerical results for comparisons. Classical Air–SF$$_{6}$$6 and Air–Helium conditions are applied in this study, and initial amplitude and Atwood number are varied for comparison. Comparisons to the simulation results from the literature show the applicability of $$\hbox {MuSiC}^{+}$$MuSiC+ and CCGF. Comparisons to the nonlinear theoretical models show that ZS (Zhang and Sohn in Phys Lett A 212:149–155, 1996; Phys Fluids 9:1106–1124, 1997), SEA (Sadot et al. in Phys Rev Lett 80:1654–1657, 1998), and DR (Dimonte and Ramaprabhu in Phys Fluids 22:014104, 2010) models are valid for both spike and bubble growth rates, and MIK (Mikaelian in Phys Rev E 67:026319, 2003) and ZG (Zhang and Guo in J Fluid Mech 786:47–61, 2016) models are valid for bubble growth rate, when the initial perturbation is small and the Atwood number is low, but only the DR model is applicable for both spike and bubble growth rates when the initial perturbation amplitude and the Atwood number are large. A new term of non-dimensional initial perturbation amplitude is presented and multiplied to the DR model to get a unified fitted DR model, which gives consistent results to the simulation ones for small and large initial amplitudes.