  In this work, we present the Domain of Dependence (DoD) stabilization for
systems of hyperbolic conservation laws in one space dimension. The base scheme
uses a method of lines approach consisting of a discontinuous Galerkin scheme
in space and an explicit strong stability preserving Runge-Kutta scheme in
time. When applied on a cut cell mesh with a time step length that is
appropriate for the size of the larger background cells, one encounters
stability issues. The DoD stabilization consists of penalty terms that are
designed to address these problems by redistributing mass between the inflow
and outflow neighbors of small cut cells in a physical way. For piecewise
constant polynomials in space and explicit Euler in time, the stabilized scheme
is monotone for scalar problems. For higher polynomial degrees $p$, our
numerical experiments show convergence orders of $p+1$ for smooth flow and
robust behavior in the presence of shocks.
