  We present a novel computational scheme to solve acoustic wave transmission
problems over composite scatterers, i.e. penetrable obstacles possessing
junctions or triple points. Our continuous problem is cast as a multiple traces
time-domain boundary integral formulation valid in two and three dimensions.
Numerically, our two-dimensional non-conforming spatial discretization uses
spectral elements based on second kind Chebyshev polynomials while a
convolution quadrature scheme is performed in the complex frequency domain.
Computational experiments reveal multistep and multistage convolution
quadrature expected convergence results for a variety of complex domains.
