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On the accuracy and stability of the perfectly matched layer in transient waveguides
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för beräkningsvetenskap. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för beräkningsvetenskap. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
2012 (Engelska)Ingår i: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 53, s. 642-671Artikel i tidskrift (Refereegranskat) Published
Ort, förlag, år, upplaga, sidor
2012. Vol. 53, s. 642-671
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
URN: urn:nbn:se:uu:diva-172993DOI: 10.1007/s10915-012-9594-7ISI: 000311400300008OAI: oai:DiVA.org:uu-172993DiVA, id: diva2:516160
Tillgänglig från: 2012-04-23 Skapad: 2012-04-17 Senast uppdaterad: 2017-12-07Bibliografiskt granskad
Ingår i avhandling
1. Perfectly Matched Layers and High Order Difference Methods for Wave Equations
Öppna denna publikation i ny flik eller fönster >>Perfectly Matched Layers and High Order Difference Methods for Wave Equations
2012 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

The perfectly matched layer (PML) is a novel technique to simulate the absorption of waves in unbounded domains. The underlying equations are often a system of second order hyperbolic partial differential equations. In the numerical treatment, second order systems are often rewritten and solved as first order systems. There are several benefits with solving the equations in second order formulation, though. However, while the theory and numerical methods for first order hyperbolic systems are well developed, numerical techniques to solve second order hyperbolic systems are less complete.

We construct a strongly well-posed PML for second order systems in two space dimensions, focusing on the equations of linear elasto-dynamics. In the continuous setting, the stability of both first order and second order formulations are linearly equivalent. We have found that if the so-called geometric stability condition is violated, approximating the first order PML with standard central differences leads to a high frequency instability at most resolutions. In the second order setting growth occurs only if growing modes are well resolved. We determine the number of grid points that can be used in the PML to ensure a discretely stable PML, for several anisotropic elastic materials.

We study the stability of the PML for problems where physical boundaries are important. First, we consider the PML in a waveguide governed by the scalar wave equation. To ensure the accuracy and the stability of the discrete PML, we derived a set of equivalent boundary conditions. Second, we consider the PML for second order symmetric hyperbolic systems on a half-plane. For a class of stable boundary conditions, we derive transformed boundary conditions and prove the stability of the corresponding half-plane problem. Third, we extend the stability analysis to rectangular elastic waveguides, and demonstrate the stability of the discrete PML.

Building on high order summation-by-parts operators, we derive high order accurate and strictly stable finite difference approximations for second order time-dependent hyperbolic systems on bounded domains. Natural and mixed boundary conditions are imposed weakly using the simultaneous approximation term method. Dirichlet boundary conditions are imposed strongly by injection. By constructing continuous strict energy estimates and analogous discrete strict energy estimates, we prove strict stability.

Ort, förlag, år, upplaga, sidor
Uppsala: Acta Universitatis Upsaliensis, 2012. s. 47
Serie
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 931
Nyckelord
Elastic waves, Surface waves, Perfectly matched layers, High order difference methods, Stability, Summation-by-parts operators, Boundary treatments
Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
Beräkningsvetenskap med inriktning mot numerisk analys
Identifikatorer
urn:nbn:se:uu:diva-173009 (URN)978-91-554-8365-4 (ISBN)
Disputation
2012-06-08, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:00 (Engelska)
Opponent
Handledare
Forskningsfinansiär
Vetenskapsrådet, VR 2009-5852
Tillgänglig från: 2012-05-14 Skapad: 2012-04-17 Senast uppdaterad: 2012-10-05Bibliografiskt granskad

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Duru, KennethKreiss, Gunilla

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