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We develop a family of cut finite element methods of different orders basedon the discontinuous Galerkin framework, for hyperbolic conservation laws withstationary interfaces in both one and two space dimensions, and for movinginterfaces in one space dimension. Interface conditions are imposed weakly andso that both conservation and stability are ensured. A CutFEM withdiscontinuous elements in space is developed and coupled to standard explicittime-stepping schemes for linear advection problems and the acoustic waveproblem with stationary interfaces. In the case of moving interfaces, wepropose a space-time CutFEM based on discontinuous elements both in space andtime for linear advection problems. We show that the proposed CutFEM areconservative and energy stable. For the stationary interface case an a priorierror estimate is proven. Numerical computations in both one and two spacedimensions support the analysis, and in addition demonstrate that the proposedmethods have the expected accuracy.

High order discontinuous cut finite element methods for linear hyperbolic conservation laws with an interface