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

TL;DR

This paper introduces a seamless extended Discontinuous Galerkin method for advection-diffusion problems on unbounded domains, combining polynomial and Laguerre basis functions for stability and efficiency in simulating fluid flows.

Contribution

The paper develops a novel extended DG discretization that couples finite and semi-infinite domains using different basis functions, with proven stability and effective boundary absorption.

Findings

Unconditional stability with respect to Péclet number.

Negligible errors from basis function mismatch.

Efficient simulation of outgoing waves with few modes.

Abstract

We propose and analyze a seamless extended Discontinuous Galerkin (DG) discretization of advection-diffusion 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 unconditional stability with respect to 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 more accurately than standard approaches at a given computational cost, thus providing an appealing model for fluid flow simulations in unbounded regions.