  In this paper, we develop a family of high order cut discontinuous Galerkin
(DG) methods for hyperbolic conservation laws in one space dimension. The ghost
penalty stabilization is used to stabilize the scheme for small cut elements.
The analysis shows that our proposed methods have similar stability and
accuracy properties as the standard DG methods on a regular mesh. We also prove
that the cut DG method with piecewise constants in space is total variation
diminishing (TVD). We use the strong stability preserving Runge-Kutta method
for time discretization and the time step is independent of the size of cut
element. Numerical examples demonstrate that the cut DG methods are high order
accurate for smooth problems and perform well for discontinuous problems.
