The time-dependent Schr"{o}dinger equation is used to study the formation of charmonium in heavy ion collisions and its propagation in Quark-Gluon Plasma (QGP) and free space. The initial bound (ground) state is computed using imaginary-time propagation in a confining potential. The QGP is simulated with a confining potential of an extended asymptotic-freedom region. The initial state propagates in real time but the charmonium bound state may become fully developed before or after the QGP formation. To this end the formation-time scales determine the kind of potential in which the quark-antiquark pair propagates and may necessitate the introduction of dissociating potentials. The survival probability is calculated versus time for various potential parameters and relative momenta of the charmonium by projecting the interacting wavefunction onto its freely-propagating counterpart. In these calculations the staggered-leap frog method is used with special attention paid to the issue of stability. It is found that charmonium decay is typically non-exponential. Fast moving states Connection with experimental results is done by means of cross-section ratios. It is shown that suppression and enhancement are both possible depending on the time-scales.