  We develop a family of cut finite element methods of different orders based
on the discontinuous Galerkin framework, for hyperbolic conservation laws with
stationary interfaces in both one and two space dimensions, and for moving
interfaces in one space dimension. Interface conditions are imposed weakly and
so that both conservation and stability are ensured. A CutFEM with
discontinuous elements in space is developed and coupled to standard explicit
time-stepping schemes for linear advection problems and the acoustic wave
problem with stationary interfaces. In the case of moving interfaces, we
propose a space-time CutFEM based on discontinuous elements both in space and
time for linear advection problems. We show that the proposed CutFEM are
conservative and energy stable. For the stationary interface case an a priori
error estimate is proven. Numerical computations in both one and two space
dimensions support the analysis, and in addition demonstrate that the proposed
methods have the expected accuracy.
