The presence of a stable density stratification is the fundamental property of both the atmosphere and natural bodies of water on scales ranging from those associated with small-scale turbulence to those large enough so as to be affected by the Earth's rotation. In this talk I will discuss the numerical simulation of stratified fluid dynamics with a focus on internal wave processes. I will describe the benefits of high-order methods, both for purely numerical simulation and for instances where it is coupled with semi-analytical theory to derive new results. In particular I will discuss fully nonlinear trapped waves over topography, the instability of the bottom boundary layer beneath internal solitary waves and the weakly non-hydrostatic dynamics of small to mid-sized lakes such as those typically found on the Canadian Shield. Throughout, I will introduce the necessary technical vocabulary and will attempt to explain the reasons for the various mathematical developments.