Background: Experimental data from heavy-ion experiments at RHIC-BNL and LHC-CERN are quantitatively described using relativistic fluid dynamics. Even p+A and p+p collisions show signs of collective behavior describable in the same manner. Nevertheless, small system sizes and large gradients strain the limits of applicability of fluid-dynamical methods. Purpose: The range of applicability of fluid dynamics for the description of the collective behavior, and in particular of the elliptic flow, of small systems needs to be explored. Method: Results of relativistic fluid-dynamical simulations are compared with solutions of the Boltzmann equation in a longitudinally boost-invariant picture. As the initial condition, several different transverse energy-density profiles for equilibrated matter are investigated. Results: While there is overall a fair agreement of energy- and particle-density profiles, components of the shear-stress tensor are more sensitive to details of the implementation. The highest sensitivity is exhibited by quantities influenced by properties of the medium at freeze-out. Conclusions: For some quantities, like the shear-stress tensor, agreement between fluid dynamics and transport theory extends into regions of Knudsen numbers and inverse Reynolds numbers where relativistic fluid dynamics is believed to fail.