Abstract. In many plasma physical and astrophysical problems, both linear and nonlinear effects can lead to global dynamics that induce, or occur simultaneously with, local phenomena. For example, a magnetically confined plasma column can potentially posses global magnetohydrodynamic (MHD) eigenmodes with an oscillation frequency that matches a local eigenfrequency at some specific internal radius. The corresponding linear eigenfunctions then demonstrate large-scale perturbations together with fine-scale resonant behaviour. A well-known nonlinear effect is the steepening of waves into shocks where the discontinuities that then develop can be viewed as extreme cases of ‘short wavelength ’ features. Numerical simulations of these types of physics problems can benefit greatly from dynamically controlled grid adaptation schemes. Here, we present a progress report on two different approaches that we envisage to evaluate against each other and use in multi-dimensional hydro- and magnetohydrodynamic computations. In r-refinement, the number of grid points stays fixed, but the grid ‘moves ’ in response to persistent or developing steep gradients. First results on 1D and 2D MHD model problems are presented. In h-refinement, the resolution is raised locally without moving individual mesh points. We show 2D hydrodynamic ‘shock tube ’ evolutions where hierarchically nested patches of subsequently finer grid spacing are created and destroyed when needed. This adaptive mesh refinement technique will be further implemented in the Versatile Advection Code, so that its functionality carries over to any set of near conservation laws in one, two, or three space dimensions.