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.