As part of an international intercomparison project, a set of single-column models (SCMs) and cloud-resolving models (CRMs) are run under the weak-temperature gradient (WTG) method and the damped gravity wave (DGW) method. For each model, the implementation of the WTG or DGW method involves a simulated column which is coupled to a reference state defined with profiles obtained from the same model in radiative-convective equilibrium. The simulated column has the same surface conditions as the reference state and is initialized with profiles from the reference state. We performed systematic comparison of the behavior of different models under a consistent implementation of the WTG method and the DGW method and systematic comparison of the WTG and DGW methods in models with different physics and numerics. CRMs and SCMs produce a variety of behaviors under both WTG and DGW methods. Some of the models reproduce the reference state while others sustain a large-scale circulation which results in either substantially lower or higher precipitation compared to the value of the reference state. CRMs show a fairly linear relationship between precipitation and circulation strength. SCMs display a wider range of behaviors than CRMs. Some SCMs under the WTG method produce zero precipitation. Within an individual SCM, a DGW simulation and a corresponding WTG simulation can produce different signed circulation. When initialized with a dry troposphere, DGW simulations always result in a precipitating equilibrium state. The greatest sensitivities to the initial moisture conditions occur for multiple stable equilibria in some WTG simulations, corresponding to either a dry equilibrium state when initialized as dry or a precipitating equilibrium state when initialized as moist. Multiple equilibria are seen in more WTG simulations for higher SST. In some models, the existence of multiple equilibria is sensitive to some parameters in the WTG calculations.