Algorithms in Scientific Computing II Case Study: Shallow Water Equations

In the following case study, we consider the so-called shallow water equations. They describe the behaviour of a fluid, in particular water, of a certain (possibly varying) depth h in a twodimensional domain – imagine, for example, a puddle of water or a shallow pond (and compare the 1D sketch given in Figure 1). The main modelling assumption is that we can neglect effects of flow in vertical direction. The resulting model therefore proves to be useful in surprisingly many situations. The simulation of tsunamis, for example, can be efficiently done using shallow water equations (with appropriate extensions). While an ocean can hardly be considered as “shallow” in the usual sense, tsunami waves (in contrast to regular waves induced by wind, e.g.) affect the entire water column, such that effects of vertical flow can again be neglected. To allow for a non-even sea bottom (as required for accurate modelling of tsunamis), we include the elevation b of the sea floor in our model (compare Figure 1).