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We propose and analyze a seamless extended Discontinuous Galerkin (DG)discretization of advection-diffusion equations on semi-infinite domains. Thesemi-infinite half line is split into a finite subdomain where the model uses astandard polynomial basis, and a semi-unbounded subdomain where scaled Laguerrefunctions are employed as basis and test functions. Numerical fluxes enable thecoupling at the interface between the two subdomains in the same way asstandard single domain DG interelement fluxes. A novel linear analysis on theextended DG model yields unconditional stability with respect to the P\'ecletnumber. Errors due to the use of different sets of basis functions on differentportions of the domain are negligible, as highlighted in numerical experimentswith the linear advection-diffusion and viscous Burgers' equations. With anadded damping term on the semi-infinite subdomain, the extended framework isable to efficiently simulate absorbing boundary conditions without additionalconditions at the interface. A few modes in the semi-infinite subdomain arefound to suffice to deal with outgoing single wave and wave train signals moreaccurately than standard approaches at a given computational cost, thusproviding an appealing model for fluid flow simulations in unbounded regions.

A seamless, extended DG approach for advection-diffusion problems on unbounded domains