Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522 Finite Difference and Discontinuous Galerkin Methods for Wave Equations SIYANG WANG ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-554-9927-3 UPPSALA urn:nbn:se:uu:diva-320614 2017 Dissertation presented at Uppsala University to be publicly examined in Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, Tuesday, 13 June 2017 at 10:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Thomas Hagstrom (Department of Mathematics, Southern Methodist University). Abstract Wang, S. 2017. Finite Difference and Discontinuous Galerkin Methods for Wave Equations. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522. 53 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9927-3. Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. We construct high order finite difference methods for wave propagation in heterogeneous media with complex geometries. In addition, we derive error estimates to a class of finite difference operators applied to the acoustic wave equation. The discontinuous Galerkin method is flexible with complex geometries. Moreover, the discontinuous nature between elements makes the method suitable for multiphysics problems. We use an energy based discontinuous Galerkin method to solve a coupled acoustic-elastic problem. Keywords: Wave propagation, Finite difference method, Discontinuous Galerkin method, Stability, Accuracy, Summation by parts, Normal mode analysis Siyang Wang, Department of Information Technology, Division of Scientific Computing, Box 337, Uppsala University, SE-751 05 Uppsala, Sweden. © Siyang Wang 2017 ISSN 1651-6214 ISBN 978-91-554-9927-3 urn:nbn:se:uu:diva-320614 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-320614) Dedicated to TDB List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. IS.WANG,K.VIRTA AND G. KREISS, High order finite difference methods for the wave equation with non–conforming grid interfaces,J. Sci. Comput., 68(2016), pp 1002–1028. Contribution: S. Wang performed the analysis, most experiments, and wrote the manuscript except Section 5.3. K. Virta wrote Section 5.3 and performed the experiments therein. The ideas were developed in close collaboration between all authors. II S. WANG, An improved high order finite difference method for non–conforming grid interfaces for the wave equation,inreview, available as preprint arXiv:1702.02056. Contribution: S. Wang is the sole author of the manuscript. III S. WANG AND G. KREISS, Convergence of summation–by–parts finite difference methods for the wave equation, J. Sci. Comput., 71(2017), pp. 219–245. Contribution: S. Wang performed the analysis and experiments. The manuscript was written by S. Wang with significant input from G. Kreiss. The ideas were developed in close collaboration between both authors. IV S. WANG,A.NISSEN AND G. KREISS, Convergence of finite difference methods for the wave equation in two space dimensions,in review, available as preprint arXiv:1702.01383. Contribution: S. Wang took the lead in performing the analysis, designing the experiments, and writing the manuscript, with significant input from A. Nissen and G. Kreiss. The ideas were developed in close collaboration between all authors. VD.APPELÖ AND S. WANG, An energy based discontinuous Galerkin method for acoustic-elastic waves, technical report. Contribution: The authors derived the method together. Software implementation of the method and writing of the manuscript were carried out in a collaborative spirit with D. Appelö taking the lead. Reprints were made with permission from the publishers. Contents 1 Introduction .................................................................................................. 9 2 Acoustic and elastic waves ........................................................................ 11 2.1 Acoustic waves ............................................................................... 11 2.1.1 Wave propagation ............................................................ 11 2.1.2 The acoustic wave equation ............................................ 13 2.1.3 Boundaries and interfaces ............................................... 14 2.2 Elastic waves .................................................................................. 16 2.2.1 Hooke’s law ..................................................................... 16 2.2.2 The elastic wave equation ............................................... 17 2.2.3 Boundaries and interfaces ............................................... 19 2.3 Numerical methods ........................................................................ 20 3 The summation–by–parts finite difference method ................................. 27 3.1 Stability analysis ............................................................................ 30 3.2 Accuracy analysis ........................................................................... 34 4 The energy–based discontinuous Galerkin method ................................. 39 5 Outlook ....................................................................................................... 42 6 Summary in Swedish ................................................................................. 43 7 Acknowledgement ..................................................................................... 45 References ........................................................................................................ 47 1. Introduction The fascinating subject of wave motion finds practical applications in a wide range of science and industry. In fact, waves are integrated in our life. Dating back to pre–modern period, long–distant communications rely on, for exam- ple, sending audio messages via loud horns, an application of acoustic wave propagation. Thanks to electromagnetic waves used in telecommunication, nowadays multimedia information can be transported instantly. Waves can also have negative effects. Seismic waves created by a sudden release of en- ergy in the earth result in earthquakes, which cause damages to humans and society. These examples represent the motivation of studies of wave motion. Waves can be distinguished by their characteristics. For example, based on the ability to transport energy through a vacuum, waves can be categorized into mechanical waves and electromagnetic waves. Mechanical waves can only propagate through a medium, whereas propagation of an electromagnetic wave does not require a medium. Sound and light are familiar examples of mechanical waves and electromagnetic waves, respectively. In this thesis, we study two important mechanical waves: acoustic waves in gases or fluids, and elastic waves in solids. Mathematically, wave propagation is modeled by partial differential equa- tions, with media properties described by coefficients in the governing equa- tions. In an ideal environment when waves propagate in a homogeneous un- bounded domain, analytical solutions to the governing equations can be de- rived. However, real–world applications provide inhomogeneities, disconti- nuities, and obstacles with complex geometries. These factors make wave propagation much more complicated, as reflections, refractions and mode con- versions occur. As a consequence, solutions to the governing equations can be written by means of analytic expressions only in very special cases. For problems in more general settings, we seek accurate approximations of exact solutions by using numerical methods. The numerical methods are then im- plemented on modern computers to provide numerical simulations to improve our understanding of wave propagation, and to answer important questions in science and technology. This thesis has contributions to two numerical methods: finite difference methods and discontinuous Galerkin methods. Finite difference methods are efficient in solving wave propagation problems on rectangular–shaped do- mains with Cartesian grids, but have difficulties in handling complex geome- tries and material discontinuities. To overcome these difficulties, we have developed a finite difference method for acoustic wave propagation in inho- mogeneous media with complex geometries. The computational domain is 9 divided into subdomains at material discontinuities, and the subdomains are coupled numerically by interpolation. Geometric features at boundaries