Thomas Richter, Universität Heidelberg, Germany Eulerian formulations for fluid-structure interaction problems have the benefit, that large deformation and contact is not a limit. In contrast to the Arbitrary Lagrangian Eulerian formulation, no artificial coordinate transformation is involved. Instead, all calculations are carried out on a fixed Eulerian background mesh. This casts the method into the frame of front-capturing techniques. The interface between fluid and solid will cut through mesh elements. Furthermore, the interface is not fixed, but it will move over time. We will show, that the motion of the interface through the domain will give rise to severe numerical problems, if we aim at getting high order convergence in space and time. First, as an interface problem, the spatial discretization must take care of discontinuities at the interface. This can be achieved by a modification of the finite elements spaces (e.g. XFEM or fitted finite elements). Second, the motion of the interface from time-step to time-step leads to a deterioration of temporal accuracy. Even though similar problems appear for all Eulerian models with moving interfaces (like levelset-formulations of multiphase flows), this problem is still insufficiently discussed in literature. In this talk, we focus on the temporal discretization. We will present a high order spatial and temporal discretization scheme for Eulerian models with sharp and moving interfaces. The discretization is based on a space-time Galerkin approach that considers both the spatial and temporal motion of the domains.