Start-up flow of an Upper Convected Maxwell (UCM) fluid over a disk set in rotation is studied by using analytical and numerical methods. Both instantaneous acceleration and linear ramping are considered. A power series expansion is used to calculate velocity and stress analytically at short times and for moderate and large Deborah numbers (De). Then momentum transfer occurs in the form of a sharp propagating wave front. For small De values, a solution using singular perturbations is developed and the wave front is totally dispersed from the beginning of rotation. The numerical calculations are carried out using either standard Finite Differences (FD) or the Split Coefficient Matrix (SCM) method. The latter uses information propagating along the characteristics direction for proper treatment of the spatial derivatives. SCM is found to be more stable and efficient than FD. De values as large as 100 have been used without breakdown of the numerical scheme.