We present a solver for computing time-dependent viscoelastic flow in two space dimensions for UCM, Oldroyd-B and Giesekus fluids. The method is based on second-order accurate finite differences on composite overlapping grids, created with the grid generating program CMPGRD, which allows very complex geometries. Several numerical results are presented. We demonstrate transient viscoelastic instabilities in a channel flow with a cylinder. A study of a 2:1-contraction flow indicates why this geometry has been difficult to simulate. An example of a 4:1-contraction shows that the Giesekus model does give rise to vortices very similar to the so-called lip vortices observed in experiments. We also give an industrial application by demonstrating the elastic effects in a paper-coating flow.