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

TL;DR

This paper presents an extended discontinuous Galerkin method for efficiently solving hyperbolic-parabolic problems on multi-dimensional unbounded domains, combining bounded finite elements with Laguerre functions for the semi-infinite part.

Contribution

It introduces a novel coupling strategy using Legendre and Laguerre basis functions for semi-infinite domains, enabling accurate and cost-effective simulations of large-scale transient dynamics.

Findings

Accurate solutions for linear and nonlinear problems.

Effective damping of outgoing signals with minimal reflections.

Significant reduction in computational cost compared to traditional methods.

Abstract

We introduce an extended discontinuous Galerkin discretization of hyperbolic-parabolic problems on multidimensional semi-infinite domains. Building on previous work on the one-dimensional case, we split the strip-shaped computational domain into a bounded region, discretized by means of discontinuous finite elements using Legendre basis functions, and an unbounded subdomain, where scaled Laguerre functions are used as a basis. Numerical fluxes at the interface allow for a seamless coupling of the two regions. The resulting coupling strategy is shown to produce accurate numerical solutions in tests on both linear and non-linear scalar and vectorial model problems. In addition, an efficient absorbing layer can be simulated in the semi-infinite part of the domain in order to damp outgoing signals with negligible spurious reflections at the interface. By tuning the scaling parameter of the Laguerre basis functions, the extended DG scheme simulates transient dynamics over large spatial scales with a substantial reduction in computational cost at a given accuracy level compared to standard single-domain discontinuous finite element techniques.