Trefftz Discontinuous Galerkin approximation of an acoustic waveguide

TL;DR

This work extends Trefftz Discontinuous Galerkin methods to time-harmonic acoustic scattering in an infinite waveguide with a potentially absorbing scatterer by introducing a modified TDG formulation that is compatible with lossy media. A Neumann-to-Dirichlet map on truncation boundaries substitutes the DtN approach to impose radiation conditions in a bounded domain, and a novel flux-based TDG scheme achieves a consistent and coercive variational formulation. The authors prove h- and p-convergence in the L^2 norm for plane-wave discretizations and validate the theory through numerical experiments, including fundamental solutions and lossy scatterers, highlighting benefits and limitations in mesh refinement and plane-wave counts. The results provide a practical framework for efficient high-frequency waveguide simulations and establish groundwork for inverse scattering data generation, with future directions including complex-n analysis and enhanced ill-conditioning mitigation. The methodology blends NtD termination, loss-aware TDG formulation, and plane-wave discretizations to handle propagating and evanescent modes in waveguides.

Abstract

We propose a modified Trefftz Discontinuous Galerkin (TDG) method for approximating a time-harmonic acoustic scattering problem in an infinitely elongated waveguide. In the waveguide we suppose there is a bounded, penetrable and possibly absorbing scatterer. The classical TDG is not applicable to this important case. Novel features of our modified TDG method are that it is applicable when the scatterer is absorbing, and it uses a stable treatment of the asymptotic radiation condition for the scattered field. For the modified TDG, we prove $h$ and $p$-convergence in the $L^2$ norm. The theoretical results are verified numerically for a discretization based on plane waves (that may be evanescent in the scatterer).

Figures (8)

• Figure 1: A sketch of the geometry of the model problem.
• Figure 2: Convergence measured in the relative $L^2(\Omega_R)$ norm as a function of the discretization parameters. Top panel: convergence as $N_p$, the number of plane waves per element, increases for various fixed meshes. Lower panel: convergence as the mesh diameter $h$ decreases using different numbers of directions per element. Here the approximation of the fundamental solution $G^{N_f}_k(\boldsymbol{x},\boldsymbol{y})$ is used as the incident field, and no scatterer is present.
• Figure 3: Relative $L^2(\Omega_R)$ error in approximating $G^{N_f}_k(\boldsymbol{x},\boldsymbol{y})$ as a function of the truncation parameter $M$ for the NtD map at several wave numbers. It is necessary to include at least $\left\lfloor \frac{k\, H}{\pi}\right\rfloor$ modes (i.e. all propagating modes), and the inclusion of a few evanescent modes results in a rapid decrease in the relative error.
• Figure 4: A density plot of the magnitude of the total field computed by TDG when $kh=6.67\times 10^{-1}$ and $N_p=11$ on $\Omega_R$ for $R=1$ when the scatterer is absorbing. The black square is the boundary of the scatterer where $\mathrm{n}=9+4i$, and the mesh is refined inside that region.
• Figure 5: Convergence curves for our TDG method with the lossy penetrable scatterer described in Subsection \ref{['scatter']}. As shown in Fig. \ref{['fig:field_absorbing']} the mesh inside the scatterer is refined slightly. The top panel shows the relative $L^2(\Omega_R)$ error as the number of plane waves $N_p$ is increased. The lower panel shows the relative $L^2(\Omega_R)$ error as the mesh is refined.

Theorems & Definitions (12)

• Lemma 1
• Remark 1
• Lemma 2
• proof
• Lemma 3
• Lemma 4
• Remark 2
• Lemma 5
• Theorem 1
• Lemma 6