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We consider spectral discretizations of hyperbolic problems on unboundeddomains using Laguerre basis functions. Taking as model problem the scalaradvection equation, we perform a comprehensive stability analysis that includesstrong collocation formulations, nodal and modal weak formulations, with eitherinflow or outflow boundary conditions, using either Gauss - Laguerre or Gauss -Laguerre - Radau quadrature nodes and based on either scaled Laguerre functionsor scaled Laguerre polynomials. We show that some of these combinations giverise to intrinsically unstable schemes, while the combination of scaledLaguerre functions with Gauss - Laguerre - Radau nodes appears to be stable forboth strong and weak formulations. We then show how a modal discretizationapproach for hyperbolic systems on an unbounded domain can be naturally andseamlessly coupled to a discontinuous finite element discretization on a finitedomain. Examples of one dimensional hyperbolic systems are solved with theproposed domain decomposition technique. The errors obtained with the proposedapproach are found to be small, enabling the use of the coupled scheme for thesimulation of Rayleigh damping layers in the semi-infinite part. Energy errorsand reflection ratios of the scheme in absorbing wavetrains and single Gaussiansignals show that a small number of nodes in the semi-infinite domain aresufficient to damp the waves. The theoretical insight and numerical resultscorroborate previous findings by the authors and establish the scaled Laguerrefunctions-based discretization as a flexible and efficient tool for absorbinglayers as well as for the accurate simulation of waves in unbounded regions.

A seamless extension of DG methods for hyperbolic problems to unbounded domains