A Seamless, Extended DG Approach for Advection–Diffusion Problems on Unbounded Domains

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 semiunbounded 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 Péclet 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.

Efficient hyperbolic–parabolic models on multi‐dimensional unbounded domains using an extended DG approach

“…The present study extends the previous one-dimensional proof-of-concept models [23][24][25] in two unexplored directions:…”

“…On the other hand, no perturbation leaving the finite domain should reach the last Laguerre node, so 𝛽 should be tuned in such a way that the semi-infinite D4 in Appendix D), in line with previous published work. 25 It is also worth noting that, with the current set of parameters, the number of total entries of the matrix of the XDG scheme is almost 4 times smaller than the corresponding matrix for the standalone DG scheme. For M = 10 and M = 40 Laguerre basis functions, the matrix of the XDG scheme has about half as many nonzero entries as the matrix for the standalone DG scheme.…”

“…In the second set of tests, we validate the accuracy of the coupling between the finite part and the semi-infinite part of the 2D strip, comparing the results obtained with the 2D XDG scheme and those obtained with a standalone DG discretization on a larger domain, [23][24][25] still considering the undamped case 𝛾 = 0. We remark that the current implementation of the XDG model enables the use of polynomial degree up to 4 for the finite part, as shown in the convergence tests and later in the efficiency tests of section 4.3.…”

“…and we discretize the problem using an XDG scheme. 25 In two dimensions, XDG is a coupled scheme consisting of • a standard DG scheme in both directions in the rectangle…”

“…Scaled Laguerre functions are chosen as basis functions in the z-direction. 19,[23][24][25] Defining…”