  Mitigating the impact of waves leaving a numerical domain has been a
persistent challenge in numerical modeling. Reducing wave reflection at the
domain boundary is crucial for accurate simulations. Absorbing layers, while
common, often incur significant computational costs. This paper introduces an
efficient application of a Legendre-Laguerre basis for absorbing layers for
two-dimensional non-linear compressible Euler equations. The method couples a
spectral-element bounded domain with a semi-infinite region, employing a tensor
product of Lagrange and scaled Laguerre basis functions. Semi-infinite elements
are used in the absorbing layer with Rayleigh damping. In comparison to
existing methods with similar absorbing layer extensions, this approach, a
pioneering application to the Euler equations of compressible and stratified
flows, demonstrates substantial computational savings. The study marks the
first application of semi-infinite elements to mitigate wave reflection in the
solution of the Euler equations, particularly in nonhydrostatic atmospheric
modeling. A comprehensive set of tests demonstrates the method's versatility
for general systems of conservation laws, with a focus on its effectiveness in
damping vertically propagating mountain gravity waves, a benchmark for
atmospheric models. Across all tests, the model presented in this paper
consistently exhibits notable performance improvements compared to a
traditional Rayleigh damping approach.
