The lifecycle of stars is profoundly shaped by the internal transport and mixing of chemical elements. Within most stars, regions of convective motions and stable stratification coexist, separated by so-called convective boundaries. While convective regions are very efficient at mixing chemical species, stably stratified regions mostly inhibit the vertical transport of elements. Current models suggest that mixing beyond convective regions is needed to explain observed stellar properties. Waves, excited by convection at convective boundaries, could play a crucial role by providing additional mixing in the vertical direction.

In this talk, I will highlight results from recent numerical simulations with the MUSIC code on the study of waves, convection, and mixing in stars. I will discuss challenges in measuring vertical mixing and transport by waves in numerical simulations, and present some prospects for improving our understanding of mixing mechanisms through numerical experiments.