The dynamics of geometric PDEs: Surface evolution equations and a comparison with their small gradient approximations

Apart from three-dimensional continuum and discrete models, the evolution of surfaces is usually described by spatially two-dimensional partial differential equations (PDEs). These models are often derived from or at least motivated by small gradient approximations, but the studied surfaces do not fulfill this requirement in all cases. We will investigate how to overcome the small gradient approximation by using geometric PDEs. Therefore, we will introduce a method to simulate the evolution of surfaces with respect to local geometric properties. In contrast to traditional PDEs, this method does not depend on the parametrization of the surface. It will not only allow us to simulate surface evolution on flat geometries but also on more complex shaped objects. For small gradients, the studies of simple model equations show similar results compared to the related PDEs. For large gradients the results differ fundamentally. Hence, the small gradient approximation should only be applied in cases where large gradients does not appear. Specifically, we exemplify this using various equations including the (damped) Kuramoto-Sivashinsky equation, which is used as a minimal model for low-energetic erosion and deposition processes, and its geometric PDE counterpart.