Abstract The explicitly elliptic momentum equation (EEME) formulation for the upper-convected Maxwell model is extended to the analysis of time-dependent flows. The formulation makes explicit the elliptic operator, which controls the spatial dependence of the velocity field, and the hyperbolic-like character of the time-dependent equations, which gives rise to the wave-like character of transients when inertia is included. Numerical solutions are computed by using finite element approximations for the spatial dependence of the velocity, stress and pressure, coupled with finite difference discretizations of time derivatives. Numerically stable and accurate solutions are demonstrated for the flows between concentric and eccentric rotating cylinders when the condition for change of type of the steady momentum equation from elliptic to hyperbolic is not satisfied. A semi-implicit algorithm is presented in which the normal stresses are computed implicitly from the corresponding components of the constitutive equation and the velocities, pressure and shear stress are computed implicitly from the EEME-continuity pair and the shear component of the constitutive equation at each time step. The coupling between these two calculations is explicit; however, calculations are not constrained to small time steps. An algorithm that decouples all stress components from velocities and pressure is shown to be severely limited to small time steps. These results are consistent with the constraints caused by the mathematical type of the governing equations.