A seamless extension of DG methods for hyperbolic problems to unbounded domains

TL;DR

This paper develops and analyzes spectral discretizations using Laguerre basis functions for hyperbolic problems on unbounded domains, demonstrating stability and effectiveness in wave absorption and simulation.

Contribution

It introduces a stable combination of scaled Laguerre functions with Gauss-Laguerre-Radau nodes for hyperbolic problems on unbounded domains and couples it with finite element methods for practical simulations.

Findings

The combination of scaled Laguerre functions and Gauss-Laguerre-Radau nodes is stable.

Small semi-infinite domain nodes effectively damp waves in simulations.

The method accurately simulates wave propagation and absorption in unbounded regions.

Abstract

We consider spectral discretizations of hyperbolic problems on unbounded domains using Laguerre basis functions. Taking as model problem the scalar advection equation, we perform a comprehensive stability analysis that includes strong collocation formulations, nodal and modal weak formulations, with either inflow or outflow boundary conditions, using either Gauss - Laguerre or Gauss - Laguerre - Radau quadrature nodes and based on either scaled Laguerre functions or scaled Laguerre polynomials. We show that some of these combinations give rise to intrinsically unstable schemes, while the combination of scaled Laguerre functions with Gauss - Laguerre - Radau nodes appears to be stable for both strong and weak formulations. We then show how a modal discretization approach for hyperbolic systems on an unbounded domain can be naturally and seamlessly coupled to a discontinuous finite element discretization on a finite domain. Examples of one dimensional hyperbolic systems are solved with the proposed domain decomposition technique. The errors obtained with the proposed approach are found to be small, enabling the use of the coupled scheme for the simulation of Rayleigh damping layers in the semi-infinite part. Energy errors and reflection ratios of the scheme in absorbing wavetrains and single Gaussian signals show that a small number of nodes in the semi-infinite domain are sufficient to damp the waves. The theoretical insight and numerical results corroborate previous findings by the authors and establish the scaled Laguerre functions-based discretization as a flexible and efficient tool for absorbing layers as well as for the accurate simulation of waves in unbounded regions.