Simple circular waveguides promise to be an ideal architecture for building high-precision matter-wave interferometers that exploit the coherent source of atoms provided by Bose-Einstein condensates (BECs). Using finite difference methods, we perform numerical calculations of the time-dependent Gross-Pitaevskii equation in one and two dimensions to simulate gravity-induced quantum interference for counterpropagating BECs in a circular waveguide. The aim of this work is to clearly understand the impact multimode excitations and nonlinear interactions have on the feasibility of interferometric measurements. Our results vividly illustrate many of the challenges to be expected in performing these types of experiments.