  We consider spectral discretizations of hyperbolic problems on unbounded
domains using Laguerre basis functions. Taking as model problem the scalar
advection equation, we perform a comprehensive stability analysis that includes
strong collocation formulations, nodal and modal weak formulations, with either
inflow or outflow boundary conditions, using either Gauss - Laguerre or Gauss -
Laguerre - Radau quadrature nodes and based on either scaled Laguerre functions
or scaled Laguerre polynomials. We show that some of these combinations give
rise to intrinsically unstable schemes, while the combination of scaled
Laguerre functions with Gauss - Laguerre - Radau nodes appears to be stable for
both strong and weak formulations. We then show how a modal discretization
approach for hyperbolic systems on an unbounded domain can be naturally and
seamlessly coupled to a discontinuous finite element discretization on a finite
domain. Examples of one dimensional hyperbolic systems are solved with the
proposed domain decomposition technique. The errors obtained with the proposed
approach are found to be small, enabling the use of the coupled scheme for the
simulation of Rayleigh damping layers in the semi-infinite part. Energy errors
and reflection ratios of the scheme in absorbing wavetrains and single Gaussian
signals show that a small number of nodes in the semi-infinite domain are
sufficient to damp the waves. The theoretical insight and numerical results
corroborate previous findings by the authors and establish the scaled Laguerre
functions-based discretization as a flexible and efficient tool for absorbing
layers as well as for the accurate simulation of waves in unbounded regions.
