Local space-time adaptive methods are developed including high-order accurate nonreflecting boundary conditions (NRBC) for time-dependent waves. The time-discontinuous Galerkin (TDG) variational method is used to divide the time-interval into space-time slabs, the solution advanced from one slab to the next. Within each slab, a continuous space-time mesh is used which enables local sub-time steps. By maintaining orthogonality of the space-time mesh and pre-integrating analytically through the time-slab, we obtain an efficient yet robust local space-time adaptive method. Any standard spatial element may be used together with standard spatial mesh generation and visualization methods. Recovery based error estimates are used in both space and time dimensions to determine the number and size of local space-time elements within a global time step such that both the spatial and temporal estimated error is equally distributed throughout the space-time approximation. The result is an efficient and reliable adaptive strategy which distributes local space-time elements where needed to accurately track time-dependent waves over large distances and time. Numerical examples of time-dependent acoustic radiation are given which demonstrate the accuracy, reliability and efficiency gained from this new technology.

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