Cut cells provide efficient meshing techniques based on a simple idea: The geometry of interest is placed in a cartesian (background) mesh. Intersections of the geometry with cartesian mesh elemetns are then removed, leading to so called cut cells while the rest of the background mesh stays untouched. While simple and efficient, the cut cells in the mesh can be of arbitrary size and shape. In the context of hyperbolic conservation laws, where explicit Runge-Kutta methods are very popular, stabilization is needed to achieve a reasonable time step size that is independent of the size of small cut cells in the mesh.
By now a variety of stabilization methods have been developed that successfully address this issue. However, the question of structure-preserving properties of the resulting schemes is still delicate: Even if the stabilization is applied to a structure-preserving base scheme, it still has to be carefully constructed to not lose any desired properties.
In this contribution we will consider the Domain of Dependence stabilization and discuss how it can be applied to an entropy-preserving DG scheme without destroying entropy preservation.