Interaction between structures and the flow around them requires a correct treatment of both of them. If the computational requirements are high, the methods need to be efficient on modern supercomputers that offer a high number of compute nodes and cores. In structural mechanics, often Finite Element (FE) methods are used, while fluid dynamics (especially for compressible flows) often apply Finite Volume (FV) methods. The best of two worlds is obtained with Discontinuous Galerkin (DG) methods, which are highly appropriate in regions with discontinuous solutions, but also highly accurate in regions with smooth solutions. The variation of h and p (mesh size and order of the polynomial) gives additional freedom to adopt to modern supercomputers (h- and p-adaptation). Nevertheless, thinking about the equations, an additional parameter for adaptation is available (e-adaptation). Also adaptation in time (t-adaptation) is relevant, which needs to go beyond the classical time step adaption. Thinking of the next generation of HPC, the efficient usage of those extremely large systems requires even parallelization in time. At this point, Continuous Galerkin (CG) methods come back into play again. This presentation will introduce the interplay of application and numerical method (quality of the solution) as well as the interplay of numerical method and suitability for highly scalable compute systems (co-design), and show some examples of flow around moving geometries (represented as an immersed boundary).