  For solving unsteady hyperbolic conservation laws on cut cell meshes, the so
called small cell problem is a big issue: one would like to use a time step
that is chosen with respect to the background mesh and use the same time step
on the potentially arbitrarily small cut cells as well. For explicit time
stepping schemes this leads to instabilities. In a recent preprint
[arXiv:1906.05642], we propose penalty terms for stabilizing a DG space
discretization to overcome this issue for the unsteady linear advection
equation. The usage of the proposed stabilization terms results in stable
schemes of first and second order in one and two space dimensions. In one
dimension, for piecewise constant data in space and explicit Euler in time, the
stabilized scheme can even be shown to be monotone. In this contribution, we
will examine the conditions for monotonicity in more detail.
