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Perfectly Matched Layers and High Order Difference Methods for Wave

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Perfectly Matched Layers and High Order Difference Methods for Wave Dedicated to my father Mr. Ambrose Duru (1939 – 2001) List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I K. Duru and G. Kreiss, (2012). A Well–posed and discretely stable perfectly matched layer for elastic wave equations in second order formulation, Commun. Comput. Phys., 11, 1643–1672 (DOI:10.4208/cicp.120210.240511a). contributions: The author of this thesis initiated this project and performed all numerical experiments. The manuscript was prepared in close collaboration between the authors II G. Kreiss and K. Duru, (2012). Discrete stability of perfectly matched layers for anisotropic wave equations in first and second order formulation, (Submitted). contributions: The author of this thesis initiated this project and performed all numerical experiments. The manuscript was prepared in close collaboration between the authors. III K. Duru and G. Kreiss, (2012). On the accuracy and stability of the perfectly matched layers in transient waveguides, J. of Sc. Comput., DOI: 10.1007/s10915-012-9594-7. In press. contributions: The author of this thesis initiated this project performed all numerical experiments and had the responsibility of writing the paper. The remaining time was spent between the author and his advisor correcting misconceptions, improving the texts and the theory. IV K. Duru and G. Kreiss, (2012). Boundary waves and stability of the perfectly matched layer. Technical report 2012-007, Department of Information Technology, Uppsala University, (Submitted) contributions: The author of this thesis initiated this project and had the responsibility of writing the paper. The remaining time was spent between the author and his advisor correcting misconceptions, improving the texts and the theory. V K. Duru and G. Kreiss, (2012). Numerical interaction of boundary waves with perfectly matched layers in elastic waveguides. Technical report 2012-008, Department of Information Technology, Uppsala University, (Submitted). contributions: The author of this thesis initiated this project, performed all numerical experiments and had the responsibility of writing the paper. The remaining time was spent between the author and his advisors correcting misconceptions, improving the texts and the theory. VI K. Duru, G. Kreiss and K. Mattsson, (2012). Accurate and stable boundary treatments for elastic wave equations in second order formulation, (Submitted). contributions: The author of this thesis initiated this project, performed all numerical experiments and had the responsibility of writing the paper. The remaining time was spent between the author and his advisors correcting misconceptions, improving the texts and the theory. Reprints were made with permission from the publishers. Contents 1 Introduction .................................................................................................. 9 2 Non-reflecting boundary conditions (NRBC) .......................................... 12 2.1 Exact NRBC ................................................................................... 12 2.2 Local NRBC ................................................................................... 14 3 Absorbing layers ........................................................................................ 16 3.1 Model problem ............................................................................... 18 3.2 Construction of the PML equations .............................................. 18 3.2.1 First order formulation .................................................... 19 3.2.2 Perfect matching .............................................................. 20 3.3 Well–posedness of the PML .......................................................... 21 3.4 Stability of the Cauchy PML ......................................................... 22 3.5 Stability of the PML for IBVPs .................................................... 23 3.6 Stability of the discrete PML ........................................................ 26 4 Finite difference methods for second order systems ............................... 29 4.1 The wave equation ......................................................................... 30 4.2 Integration–by–parts ...................................................................... 31 4.3 Summation–by–parts ..................................................................... 31 4.4 Weak boundary treatment with SAT ............................................. 32 5 Summary of papers .................................................................................... 34 5.1 Paper I ............................................................................................. 34 5.2 Paper II ............................................................................................ 35 5.3 Paper III .......................................................................................... 35 5.4 Paper IV .......................................................................................... 36 5.5 Paper V ........................................................................................... 36 5.6 Paper VI .......................................................................................... 37 6 Summary in Swedish ................................................................................. 39 References ........................................................................................................ 42 1. Introduction Wave motion is a dominant feature for problems in many branches of engi- neering and applied sciences. For instance, both industrial materials such as steel pipes and plates [78, 93], and natural structures like the Earth’s surface can support propagating waves that can lead to failure or disaster. Numerical simulations of propagating waves can serve as a complement to theoretical and experimental investigations, and can possibly treat many more important sce- narios that can not be investigated by theory or experiments. Thus, providing valuable information which can be useful in developing modern technologies, exploring natural minerals from the subsurface, and understanding natural dis- asters such as earthquakes and tsunamis. A defining feature of propagating waves is that they can propagate long dis- tances relative to the characteristic dimension, the wavelength. As an example, strong ground shaking resulting from an earthquake or a nuclear explosion can be recorded far away in another continent sometime after it has occurred. In practice, because of limited resources, computer simulations of such prob- lems are restricted only to areas of interest, where the effects of the strong ground motion is significant. This is typical of numerical simulations of many wave propagation problems including seismic imaging, wireless communica- tion and ground penetrating radar (GPR) technologies. In numerical simulations, large spatial domains must be truncated to fit into the finite memory of the computer, by introducing artificial boundaries. One can immediately pose the question: Which boundary conditions ensure that the numerical solution of the initial boundary value problem inside the truncated domain converge to the solution of the original problem in the unbounded domain? Provided the numerical method used in the interior is consistent and stable, the numerical solution will converge to the solution of the original problem in the unbounded domain only if artificial boundaries are closed with ‘accu- rate’ and reliable boundary conditions. Otherwise, waves traveling out of the computational domain generate spurious reflections at the boundaries, which will then travel back into the computational domain and pollute the solution everywhere. It becomes apparent that the most important feature of artificial boundary conditions is that, all out–going waves disappear (or are absorbed) without reflections. Therefore, an effective numerical wave simulator should be able to treat both physical and artificial boundaries efficiently. 9 One objective of this thesis is to design effective artificial boundary condi- tions suitable for numerical solutions of partial differential equations (PDE) describing wave phenomena. The effort to design efficient artificial bound- ary conditions began over thirty years ago [35] and has evolved over time to become an entire area of research. Artificial boundary conditions in general can be divided into two main classes: absorbing or non-reflecting boundary conditions (NRBC) [35] and absorbing layers [20]. We also note that for time-harmonic problems there are accurate and efficient artificial boundary procedures, see [107, 47]. Problems of this class are not considered in this thesis. Most domain truncation schemes for numerical time-dependent wave prop- agation have been developed for constant coefficient wave propagation prob- lems. Many of these methods such as high order local NRBCs and the per- fectly matched layer (PML) which we will discuss in Chapter 2 and Chap- ter 3, respectively, are efficient for certain problems, particularly the scalar wave equation and the Maxwell’s equations in isotropic homogeneous me- dia. For many other problems in this class, such as the linearized magneto– hydrodynamic (MHD) equations and the equations of linear elasticity, there are yet unresolved problems, see for example [17, 11]. Artificial boundary conditions for more difficult problems such as variable coefficient and non-linear wave propagation problems are less developed. In practice, ad hoc methods are still in
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