  We introduce an extended discontinuous Galerkin discretization of
hyperbolic-parabolic problems on multidimensional semi-infinite domains.
Building on previous work on the one-dimensional case, we split the
strip-shaped computational domain into a bounded region, discretized by means
of discontinuous finite elements using Legendre basis functions, and an
unbounded subdomain, where scaled Laguerre functions are used as a basis.
Numerical fluxes at the interface allow for a seamless coupling of the two
regions. The resulting coupling strategy is shown to produce accurate numerical
solutions in tests on both linear and non-linear scalar and vectorial model
problems. In addition, an efficient absorbing layer can be simulated in the
semi-infinite part of the domain in order to damp outgoing signals with
negligible spurious reflections at the interface. By tuning the scaling
parameter of the Laguerre basis functions, the extended DG scheme simulates
transient dynamics over large spatial scales with a substantial reduction in
computational cost at a given accuracy level compared to standard single-domain
discontinuous finite element techniques.
