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  • 1.
    Duru, Kenneth
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Perfectly Matched Layers and High Order Difference Methods for Wave Equations2012Doctoral thesis, comprehensive summary (Other academic)
    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.

    List of papers
    1. A well-posed and discretely stable perfectly matched layer for elastic wave equations in second order formulation
    Open this publication in new window or tab >>A well-posed and discretely stable perfectly matched layer for elastic wave equations in second order formulation
    2012 (English)In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 11, p. 1643-1672Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-167019 (URN)10.4208/cicp.120210.240511a (DOI)000303761800009 ()
    Available from: 2012-01-12 Created: 2012-01-19 Last updated: 2017-12-08Bibliographically approved
    2. Discrete stability of perfectly matched layers for anisotropic wave equations in first and second order formulation
    Open this publication in new window or tab >>Discrete stability of perfectly matched layers for anisotropic wave equations in first and second order formulation
    2013 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 53, p. 641-663Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-172998 (URN)10.1007/s10543-013-0426-4 (DOI)000323729800005 ()
    Available from: 2013-03-19 Created: 2012-04-17 Last updated: 2017-12-07Bibliographically approved
    3. On the accuracy and stability of the perfectly matched layer in transient waveguides
    Open this publication in new window or tab >>On the accuracy and stability of the perfectly matched layer in transient waveguides
    2012 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 53, p. 642-671Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-172993 (URN)10.1007/s10915-012-9594-7 (DOI)000311400300008 ()
    Available from: 2012-04-23 Created: 2012-04-17 Last updated: 2017-12-07Bibliographically approved
    4. Boundary waves and stability of the perfectly matched layer
    Open this publication in new window or tab >>Boundary waves and stability of the perfectly matched layer
    2012 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-007
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-172887 (URN)
    Available from: 2012-04-10 Created: 2012-04-16 Last updated: 2012-05-15Bibliographically approved
    5. Numerical interaction of boundary waves with perfectly matched layers in elastic waveguides
    Open this publication in new window or tab >>Numerical interaction of boundary waves with perfectly matched layers in elastic waveguides
    2012 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-008
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-172888 (URN)
    Available from: 2012-04-11 Created: 2012-04-16 Last updated: 2012-05-15Bibliographically approved
    6. Stable and high-order accurate boundary treatments for the elastic wave equation on second-order form
    Open this publication in new window or tab >>Stable and high-order accurate boundary treatments for the elastic wave equation on second-order form
    2014 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 36, p. A2787-A2818Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-173003 (URN)10.1137/130947210 (DOI)000346838800013 ()
    Available from: 2014-12-10 Created: 2012-04-17 Last updated: 2017-12-07Bibliographically approved
  • 2.
    Duru, Kenneth
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Perfectly matched layers for second order wave equations2010Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    Numerical simulation of propagating waves in unbounded spatial domains is a challenge common to many branches of engineering and applied mathematics. Perfectly matched layers (PML) are a novel technique for simulating the absorption of waves in open domains. The equations modeling the dynamics of phenomena of interest are usually posed as differential equations (or integral equations) which must be solved at every time instant. In many application areas like general relativity, seismology and acoustics, the underlying equations are systems of second order hyperbolic partial differential equations. In numerical treatment of such problems, the equations are often rewritten as first order systems and are solved in this form. For this reason, many existing PML models have been developed for first order systems. In several studies, it has been reported that there are drawbacks with rewriting second order systems into first order systems before numerical solutions are obtained. While the theory and numerical methods for first order systems are well developed, numerical techniques to solve second order hyperbolic systems is an on-going research.

    In the first part of this thesis, we construct PML equations for systems of second order hyperbolic partial differential equations in two space dimensions, focusing on the equations of linear elasto-dynamics. One advantage of this approach is that we can choose auxiliary variables such that the PML is strongly hyperbolic, thus strongly well-posed. The second is that it requires less auxiliary variables as compared to existing first order formulations. However, in continuum the stability of both first order and second order formulations are linearly equivalent. A turning point is in numerical approximations. 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 for any given resolution. The second order discretization behaves much more stably. In the second order setting instability occurs only if unstable modes are well resolved.

    The second part of this thesis discusses the construction of PML equations for the time-dependent Schrödinger equation. From mathematical perspective, the Schrödinger equation is unique, in the sense that it is only first order in time but second order in space. However, with slight modifications, we carry over our ideas from the hyperbolic systems to the Schrödinger equations and derive a set of asymptotically stable PML equations. The new model can be viewed as a modified complex absorbing potential (CAP). The PML model can easily be adapted to existing codes developed for CAP by accurately discretizing the auxiliary variables and appending them accordingly. Numerical experiments are presented illustrating the accuracy and absorption properties of the new PML model.

    We are hopeful that the results obtained in this thesis will find useful applications in time-dependent wave scattering calculations.

    List of papers
    1. Well-posed and discretely stable perfectly matched layers for elastic wave equations in second order formulation
    Open this publication in new window or tab >>Well-posed and discretely stable perfectly matched layers for elastic wave equations in second order formulation
    2010 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2010-004
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-117577 (URN)
    Available from: 2010-02-19 Created: 2010-02-19 Last updated: 2014-06-16Bibliographically approved
    2. Discrete stability of perfectly matched layers for anisotropic wave equations in first and second order formulation
    Open this publication in new window or tab >>Discrete stability of perfectly matched layers for anisotropic wave equations in first and second order formulation
    2013 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 53, p. 641-663Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-172998 (URN)10.1007/s10543-013-0426-4 (DOI)000323729800005 ()
    Available from: 2013-03-19 Created: 2012-04-17 Last updated: 2017-12-07Bibliographically approved
    3. Stable perfectly matched layers for the Schrödinger equations
    Open this publication in new window or tab >>Stable perfectly matched layers for the Schrödinger equations
    2010 (English)In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, p. 287-295Conference paper, Published paper (Refereed)
    Place, publisher, year, edition, pages
    Berlin: Springer-Verlag, 2010
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-132926 (URN)10.1007/978-3-642-11795-4_30 (DOI)000395207900030 ()978-3-642-11794-7 (ISBN)
    Projects
    eSSENCE
    Available from: 2010-10-29 Created: 2010-10-29 Last updated: 2018-06-16Bibliographically approved
  • 3. Duru, Kenneth
    et al.
    Gabriel, Alice-Agnes
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    On energy stable discontinuous Galerkin spectral element approximations of the perfectly matched layer for the wave equation2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 350, p. 898-937Article in journal (Refereed)
  • 4. Duru, Kenneth
    et al.
    Kozdon, Jeremy E.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Boundary conditions and stability of a perfectly matched layer for the elastic wave equation in first order form2015In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 303, p. 372-395Article in journal (Refereed)
  • 5. Duru, Kenneth
    et al.
    Kozdon, Jeremy E.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Boundary waves and stability of perfectly matched layers II: Extensions to first order systems and numerical stability2013In: Proc. 11th International Conference on Mathematical and Numerical Aspects of Waves, Tunisia: ENIT , 2013, p. 301-302Conference paper (Other academic)
  • 6.
    Duru, Kenneth
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    A well-posed and discretely stable perfectly matched layer for elastic wave equations in second order formulation2012In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 11, p. 1643-1672Article in journal (Refereed)
  • 7.
    Duru, Kenneth
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Boundary waves and stability of the perfectly matched layer2012Report (Other academic)
  • 8.
    Duru, Kenneth
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Boundary waves and stability of the perfectly matched layer for the two space dimensional elastic wave equation in second order form2014In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 52, p. 2883-2904Article in journal (Refereed)
  • 9. Duru, Kenneth
    et al.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Efficient and stable perfectly matched layer for CEM2014In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 76, p. 34-47Article in journal (Refereed)
  • 10.
    Duru, Kenneth
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Numerical interaction of boundary waves with perfectly matched layers in elastic waveguides2012Report (Other academic)
  • 11.
    Duru, Kenneth
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Numerical interaction of boundary waves with perfectly matched layers in two space dimensional elastic waveguides2014In: Wave motion, ISSN 0165-2125, E-ISSN 1878-433X, Vol. 51, p. 445-465Article in journal (Refereed)
  • 12.
    Duru, Kenneth
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    On the accuracy and stability of the perfectly matched layer in transient waveguides2012In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 53, p. 642-671Article in journal (Refereed)
  • 13.
    Duru, Kenneth
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Stable perfectly matched layers for the Schrödinger equations2010In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, p. 287-295Conference paper (Refereed)
  • 14.
    Duru, Kenneth
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Well-posed and discretely stable perfectly matched layers for elastic wave equations in second order formulation2010Report (Other academic)
  • 15.
    Duru, Kenneth
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Mattsson, Ken
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Stable and high-order accurate boundary treatments for the elastic wave equation on second-order form2014In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 36, p. A2787-A2818Article in journal (Refereed)
  • 16.
    Duru, Kenneth
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Mattsson, Ken
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Stable and conservative time propagators for second order hyperbolic systems2011Report (Other academic)
  • 17. Duru, Kenneth
    et al.
    Virta, Kristoffer
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Stable and high order accurate difference methods for the elastic wave equation in discontinuous media2013In: Proc. 11th International Conference on Mathematical and Numerical Aspects of Waves, Tunisia: ENIT , 2013, p. 197-198Conference paper (Other academic)
  • 18. Duru, Kenneth
    et al.
    Virta, Kristoffer
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Stable and high order accurate difference methods for the elastic wave equation in discontinuous media2014In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 279, p. 37-62Article in journal (Refereed)
  • 19. Khan, Masood
    et al.
    Abbas, Qaisar
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Duru, Kenneth
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Magnetohydrodynamic flow of a Sisko fluid in annular pipe: A numerical study2010In: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 62, p. 1169-1180Article in journal (Refereed)
  • 20.
    Kreiss, Gunilla
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Duru, Kenneth
    Boundary waves and stability of perfectly matched layers I2013In: Proc. 11th International Conference on Mathematical and Numerical Aspects of Waves, Tunisia: ENIT , 2013, p. 299-300Conference paper (Other academic)
  • 21.
    Kreiss, Gunilla
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Duru, Kenneth
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Discrete stability of perfectly matched layers for anisotropic wave equations in first and second order formulation2013In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 53, p. 641-663Article in journal (Refereed)
1 - 21 of 21
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