Investigations of musical instruments typically carried out in the frequency domain. In contrast, numerical investigations in the time domain offer several advantages. Physical effects and single travelling waves can both be observed directly. This allows studying how connections, tone holes and bore perturbations influence the wave propagation and act as sound sources. In contrast to methods, which are formulated in the frequency domain, time domain investigations also enable the examination of transient effects. However, these are accompanied by the disadvantage that acoustic waves have to be tracked over a long period of time. When using low-order methods, numerical dissipation and dispersion errors can can have highly distortive effects on the results. In order to overcome these issues we use a high-order discontinuous Galerkin formulation. Extrapolation boundary conditions in conjunction with a slope limiting procedure provide stable, compact and non-reflecting boundary conditions. This report presents the results of numerical investigations of woodwind instruments using the examples of the recorder and the bassoon.