This project concerns numerical methods and developing novel, high-order accurate algorithms for physical models of sediment and shallow water flows. The need for computational resources comes, mainly, from the disparate time scales of these flow properties. In order to simulate relevant test cases that come from real-world applications parallel computing is a necessity. The need is two-fold. For example, modelling how the bottom of a river evolves is inherently slow and requires long time scales. The water flows in the river at what can be perceived as a "normal" rate, but the sediment is much slower so that the approximate solutions to the partial differential equations must be solved to a final time on the order of days or weeks. Additionally, to properly capture this segmentation and sediment evolution require many degrees of freedom as the domains can be quite large, e.g., the harbour of a coastal city and the nearby oceanic bathymetry.