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In this work, we present the Domain of Dependence (DoD) stabilization forsystems of hyperbolic conservation laws in one space dimension. The base schemeuses a method of lines approach consisting of a discontinuous Galerkin schemein space and an explicit strong stability preserving Runge-Kutta scheme intime. When applied on a cut cell mesh with a time step length that isappropriate for the size of the larger background cells, one encountersstability issues. The DoD stabilization consists of penalty terms that aredesigned to address these problems by redistributing mass between the inflowand outflow neighbors of small cut cells in a physical way. For piecewiseconstant polynomials in space and explicit Euler in time, the stabilized schemeis monotone for scalar problems. For higher polynomial degrees $p$, ournumerical experiments show convergence orders of $p+1$ for smooth flow androbust behavior in the presence of shocks.

DoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension