  We present a new numerical approach for wave induced dynamic fracture. The
method is based on a discontinuous Galerkin approximation of the first-order
hyperbolic system for elastic waves and a phase-field approximation of brittle
fracture driven by the maximum tension. The algorithm is staggered in time and
combines an implicit midpoint rule for the wave propagation followed by an
implicit Euler step for the phase-field evolution. At fracture, the material is
degraded, and the waves are reflected at the diffusive interfaces. Two and
three-dimensional examples demonstrate the advantages of the proposed method
for the computation of crack growth and spalling initiated by reflected and
superposed waves.
