The simplified perturbation method of Vandenboomgaerde et
al. (2002) is applied to both the
Richtmyer–Meshkov and the Rayleigh–Taylor instabilities.
This theory is devoted to the calculus of the growth rate of the
perturbation of the interface in the weakly nonlinear stage. In the
standard approach, expansions appear to be series in time. We build
accurate approximations by retaining only the terms with the highest
power in time. This simplifies and accelerates the solution. High order
expressions are then easily reachable. For the Richtmyer–Meshkov
instability, multimode configurations become tractable and the
selection mode process can be studied. Inferences for the intermediate
nonlinear regime are also proposed. In particular, a class of
homothetic configurations is inferred; its validity is verified with
numerical simulations even as vortex structures appear at the
interface. This kind of method can also be used for the
Rayleigh–Taylor instability. Some examples are presented.