The time-discontinuous Galerkin (TDG) method provides high-order accuracy with desirable C and L stability for second-order hyperbolic systems governing acoustics. C and L stability provide asymptotic annihilation of high frequency response due to any spurious resolution of small scales. In order to retain the high-order accuracy and stability properties of the parent TDG method while gaining efficiency comparable with standard second-order accurate single-step/single-solve (SS/SS) time stepping algorithms, generalised gradients of residuals of the governing equations are added to the standard TDG variational equation. Using this stabilized framework, together with optimal design of temporal approximations and time-scales, efficient multi-pass iterative solution algorithms are developed which maintain C and L stability, provide high-order accuracy in only two or three iterative passes, and can easily be implemented in standard finite element codes. An alternative decoupling strategy which uses a spectral decomposition of the time arrays is also developed and used to efficiently solve the space-time matrix equations resulting from the TDG method. This approach is especially amendable for parallel implementation. Local adaptive space-time finite element strategies based on the TDG method are also presented.

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