DMCA
An alternative to Dirichlet-to-Neumann maps for waveguides (2011)
BibTeX
@MISC{Bonnet-ben11analternative,
author = {Anne-sophie Bonnet-ben and Dhia Guillaume Legendre},
title = {An alternative to Dirichlet-to-Neumann maps for waveguides},
year = {2011}
}
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Abstract
We are interested by the treatment of the radiation condition at infinity for the numerical solution of a problem set in an unbounded waveguide. We propose an alternative to the classical approach involving a modal expression of Dirichlet-to-Neumann (DtN) operators. This method is particularly simple to implement since it only requires the solution of boundary value problems with local boundary conditions. The corresponding approximate solution is comparable in accuracy to the one obtained by truncating the infinite series in the DtN maps. 1 The diffraction problem in a semi-infinite waveguide In this note, we consider, for the sake of simplicity, an acoustic scattering problem set in a possibly locally perturbed semi-infinite waveguide. More precisely, we consider a connected unbounded do-main Ω ⊂ Rd with d ≥ 2, such that Ω∩{x = (xS, xd) |xd> 0} = {x = (xS, xd) |xS ∈ S, xd> 0}, where S is a bounded subset of Rd−1, and Ω ∩ {x = (xS, xd) |xd < 0} = Ω0 is a bounded domain (see Figure 1). We are interested in numerically solving the following boundary value problem −∆u − k2u = f in Ω, ∂nu = 0 on ∂Ω, (1) supplemented by a radiation condition at infinity. It is assumed that the source term f belongs to L2(Ω) and is compactly supported in Ω0, and that the wave number k is real. The vector n denotes the outward unit normal on ∂Ω. The prescribed radiation condition states that the solution is “outgoing ” at infinity. It can be written down by introducing the so-called modes of the guide, which are functions with separated variables of the form ϕn(xs) e ±iβnxd, n ∈ N, the complex numbers βn being such that βn =√ k2 − λn for k2 ≥ λn and βn = i λn − k2 for k2 ≤ λn. Here, the real positive scalar λn denotes the nth eigenvalue of the negative Laplace operator acting in L2(S) and associated with the homogeneous Neumann boundary condition on ∂S, and the orthonormal real-valued functions ϕn are the corresponding eigenfunctions. A mode is said to be propagative if βn ∈ R, and evanescent if βn ∈ iR (note that there is a finite number Nprop of propagative modes). If k2 6 = λn, ∀n ∈ N, S0 S`Ω0 0 ` xd Figure 1: A realization of the domain Ω. 1 ha l-0
Keyphrases
dirichlet-to-neumann map radiation condition semi-infinite waveguide finite number nprop prescribed radiation condition state so-called mode real positive scalar xd figure boundary value problem propagative mode numerical solution bounded domain complex number modal expression negative laplace operator separated variable homogeneous neumann boundary condition local boundary condition orthonormal real-valued function corresponding approximate solution acoustic scattering problem corresponding eigenfunctions unbounded waveguide ha l-0 unbounded do-main rd boundary value problem k2u bounded subset nth eigenvalue classical approach source term belongs diffraction problem dtn map wave number outward unit infinite series
