Hyperbolic evolution equations require numerical diffusion for stability (in particular when explicit time integration is used). Some essential properties of the exact solutions, however, are lost, for instance, data that should remain stationary are diffused away. In this talk I will focus on stationary states for linear acoustics and the low Mach number regime for the inviscid Euler equations. Adequate numerical solutions can in principle be obtained by grid refinement, but it easily becomes excessive and impractical. Structure preserving methods generally allow to reproduce qualitative properties of the exact solutions on coarse grds already. I will show new strategies how to achieve stationarity preservation and low Mach number compliance in Finite Volume methods, and analyze the structure preservation properties of a recently introduced method (Active Flux) that blends ideas from Finite Volume and Finite Element methods, and crucially uses continuous reconstruction only.