We simulate the Galactic Center nonthermal filaments as magnetized wakes formed dynamically from amplification of a weak (tens of mu G) global magnetic field through the interaction of molecular clouds with a Galactic Center wind. One of the key issues in this cometary model is the stability of the filament against dynamical disruption. Here we show 2-dimensional MHD simulations for interstellar conditions that are appropriate for the Galactic Center. The structures eventually disrupt through a shear driven nonlinear instability but maintain coherence for lengths up to 100 times their width as observed. The final instability, which destroys the filament through shredding and plasmoid formation, grows quickly in space (and time) and leads to an abrupt end to the structure, in accord with observations. As a by-product, the simulation shows that emission should peak well downstream from the cloud-wind interaction site.