Slow Sand Filters are used in the production of drinking water and they consist of a large filter where the sand develops a biofilm on its surface layer that acts as a biofilter. In order to model this process, a large system of non-linear PDEs is used and, due to the scale of bacterial growth on sand grains compared to the size of the filter, a very fine resolution is needed. Thus, high-performance computing is required to obtain accurate simulations over a long time period. In this project, we developed a new mathematical model and a corresponding numerical method to solve these equations. The novelty here being that, unlike previous models used in Slow Sand Filter simulation, the supernatant water and sand are not treated separately but as one connected domain with discontinuous parameters.