Absorbing boundary conditions for acoustic models at low viscosity in a waveguide. (7th March 2016)

• Record Type: Journal Article
• Authors: Semin, Adrien; Schmidt, Kersten
• Other Names: Dauge Monique guestEditor.; Nicaise Serge guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 39:Number 17(2016:Nov. 30)
• Page Start: 5043
• Page End: 5065
• Publication Date: 2016-03-07
• Subjects: acoustic equations with viscosity -- semi‐infinite waveguide -- Dirichlet‐to‐Neumann boundary conditions -- Helmholtz equation -- linearized Navier–Stokes equations; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.3755
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : 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.