Numerical models have become an indispensable tool for ocean and inland flow modeling. Such models typically use the hydrostatic approximation based on the argument that their horizontal length scales are longer than the vertical length scales. There are a wide variety of physical processes in oceans and inland water systems, and many of these processes are adequately modeled with the hydrostatic approximation. However, internal waves contribute to the physics that influence mixing in a density stratified system and have been previously shown to be non- hydrostatic. The neglect of non-hydrostatic pressure in a hydrostatic model is problematic since non-hydrostatic pressure plays a significant role in internal wave evolution balancing nonlinear wave steepening. Where non-hydrostatic pressure is neglected in a model, the governing equations are missing a piece of the physics that control the internal wave evolution, so it should not be surprising that the evolution may be poorly predicted. Despite the knowledge that the non-hydrostatic pressure is necessary for correctly modeling the physics of a steepening internal wave, the high computational cost of solving the non-hydrostatic pressure has limited its use in large-scale systems. Furthermore, the errors associated with hydrostatic modeling of internal waves have not been quantified. This dissertation quantifies the differences between hydrostatic and non-hydrostatic simulations of internal wave evolution and develops a method to a priori determine regions with non-hydrostatic behavior. In quantifying the errors and differences between the two models this research provides the characteristics of model error with grid refinement. Additionally, it is shown that hydrostatic models may develop high wavenumber “soliton-like” features that are purely a construct of model error, but may seem to mimic physical behaviors of the non-hydrostatic system. Finally, it is shown that regions of significant non-hydrostatic pressure gradients can be identified from a hydrostatic model. This latter finding is a building block towards coupling local non-hydrostatic solutions with global hydrostatic solutions for more efficient computational methods. The work presented here provides the foundations for future non-hydrostatic model development and application.