Numerical simulations of convection induced by solvent evaporation during the drying of a polymer solution are considered. This paper focuses on the transient thermal regime occurring at the beginning of the drying and transient solutal effects are not taken into account. The onset of convection (B'enard-Marangoni and Rayleigh-B'enard) is studied for a large range of initial thicknesses and viscosities. Several stochastic models are compared to analyze the influence of the perturbation description on the transition thresholds. Two-dimensional (2D) and three-dimensinal (3D) models are shown to give close results. The 3D model is used to characterize the pattern evolution during the drying. In the case of surface tension driven convection, a method is developed to describe the cells morphology and their time evolution.