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Absorbing boundary conditions for acoustic models at low viscosity in a waveguide
Author(s) -
Semin Adrien,
Schmidt Kersten
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3755
Subject(s) - mathematics , boundary value problem , viscosity , neumann boundary condition , mathematical analysis , helmholtz equation , limit (mathematics) , boundary (topology) , dirichlet boundary condition , robin boundary condition , convergence (economics) , helmholtz free energy , physics , thermodynamics , economics , economic growth
We consider different acoustic models with viscosity in a semi‐infinite waveguide with rigid walls, for which we propose and analyse absorbing boundary conditions on a truncated subdomain. The considered models are (i) the viscous acoustic equations in a stagnant mean flow, which exhibit for small viscosities boundary layers on the infinite walls, (ii) the limit equations for vanishing viscosity and (iii) a first‐order approximation for low viscosity. The limit model (i) is well known as the Helmholtz equation for the pressure with homogeneous Neumann boundary conditions. For each of these models, the absorbing conditions appear as Dirichlet‐to‐Neumann (DtN) maps. The DtN boundary conditions for the singularly perturbed model (i) and the approximative model (iii) tend to the DtN boundary conditions of the limit problem (ii) if the viscosity approaches zero, and, hence, provide a uniform accuracy in the viscosity. The convergence of truncated DtN boundary conditions and the behaviour for viscosities tending to zero are shown in numerical experiments. Copyright © 2016 John Wiley & Sons, Ltd.