  We present new stabilization terms for solving the linear transport equation
on a cut cell mesh using the discontinuous Galerkin (DG) method in two
dimensions with piecewise linear polynomials. The goal is to allow for explicit
time stepping schemes, despite the presence of cut cells. Using a method of
lines approach, we start with a standard upwind DG discretization for the
background mesh and add penalty terms that stabilize the solution on small cut
cells in a conservative way. Then, one can use explicit time stepping, even on
cut cells, with a time step length that is appropriate for the background mesh.
In one dimension, we show monotonicity of the proposed scheme for piecewise
constant polynomials and total variation diminishing in the means stability for
piecewise linear polynomials. We also present numerical results in one and two
dimensions that support our theoretical findings.
