  We propose and analyze a seamless extended Discontinuous Galerkin (DG)
discretization of hyperbolic-parabolic equations on semi-infinite domains. The
semi-infinite half line is split into a finite subdomain where the model uses a
standard polynomial basis, and a semi-unbounded subdomain where scaled Laguerre
functions are employed as basis and test functions. Numerical fluxes enable the
coupling at the interface between the two subdomains in the same way as
standard single domain DG interelement fluxes. A novel linear analysis on the
extended DG model yields stability constraints on the finite subdomain grid
size that get tighter for increasing values of the P\'eclet number. Errors due
to the use of different sets of basis functions on different portions of the
domain are negligible, as highlighted in numerical experiments with the linear
advection-diffusion and viscous Burgers' equations. With an added damping term
on the semi-infinite subdomain, the extended framework is able to efficiently
simulate absorbing boundary conditions without additional conditions at the
interface. A few modes in the semi-infinite subdomain are found to suffice to
deal with outgoing single wave and wave train signals, thus providing an
appealing model for fluid flow simulations in unbounded regions.
