Finite-difference Methods for the Wave Equation with Reduced Dispersion Errors PDF Download

Author: Yajun An
Publisher:
ISBN:
Category : Finite differences
Languages : en
Pages :

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Book Description
A new methodology was proposed in Finkelstein and Kastner (2007,2008) to derive finite-difference (FD) schemes in the joint time-space domain to reduce dispersion error. The key idea is that the true dispersion relation is satisfied exactly at some specified wavenumbers. Liu and Sen (2009) further developed their idea, going to 2D and 3D. In our work, we will prove that the system for coefficients of these new schemes is solvable for any normalized wavenumbers up to the Nyquist. We will also look at the system matrix and prove that we can get higher order approximation to the dispersion at arbitrary normalized wavenumbers up to the Nyquist.

Author: Yajun An
Publisher:
ISBN:
Category :
Languages : en
Pages : 96

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Book Description
Finite Difference (FD) schemes have been used widely in computing approximations for partial differential equations for wave propagation, as they are simple, flexible and robust. However, even for stable and accurate schemes, waves in the numerical schemes can propa- gate at different wave speeds than in the true medium. This phenomenon is called numerical dispersion error. Traditionally, FD schemes are designed by forcing accuracy conditions, and in spite of the advantages mentioned above, such schemes suffer from numerical dispersion errors. Traditionally, two ways have been used for the purpose of reducing dispersion error: increasing the sampling rate and using higher order accuracy. More recently, Finkelstein and Kastner (2007, 2008) propose a unified methodology for deriving new schemes that can accommodate arbitrary requirements for reduced phase or group velocity dispersion errors, defined over any region in the frequency domain. Such schemes are based on enforcing exact phase or group velocity at certain preset wavenumbers. This method has been shown to reduce dispersion errors at large wavenumbers. In this dissertation, we study the construction and behaviors of FD schemes designed to have reduced numerical dispersion error. We prove that the system of equations to select the coefficients in a centered FD scheme for second order wave equations with specified order of accuracy and exact phase velocity at preset wavenumbers can always be solved. Furthermore, from the existence of such schemes, we can show that schemes which reduce the dispersion error uniformly in an interval of the frequency domain can be constructed from a Remez algorithm. In these new schemes we propose, we can also specify wavenumbers where the exact phase or group dispersion relation can be satisfied. For an incoming signal consisting of waves of different wavenumbers, our schemes can give more accurate wave propagation speeds. Furthermore, when we apply our schemes in two dimensional media, we can obtain schemes that give small dispersion error at all propagation angles.

Author: Gary Cohen
Publisher: Springer Science & Business Media
ISBN: 366204823X
Category : Science
Languages : en
Pages : 355

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Book Description
"To my knowledge [this] is the first book to address specifically the use of high-order discretizations in the time domain to solve wave equations. [...] I recommend the book for its clear and cogent coverage of the material selected by its author." --Physics Today, March 2003

Author: Hans Petter Langtangen
Publisher: Springer
ISBN: 3319554565
Category : Computers
Languages : en
Pages : 522

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Book Description
This book is open access under a CC BY 4.0 license. This easy-to-read book introduces the basics of solving partial differential equations by means of finite difference methods. Unlike many of the traditional academic works on the topic, this book was written for practitioners. Accordingly, it especially addresses: the construction of finite difference schemes, formulation and implementation of algorithms, verification of implementations, analyses of physical behavior as implied by the numerical solutions, and how to apply the methods and software to solve problems in the fields of physics and biology.

Author: Dale R. Durran
Publisher: Springer Science & Business Media
ISBN: 1475730810
Category : Mathematics
Languages : en
Pages : 476

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Book Description
Covering a wide range of techniques, this book describes methods for the solution of partial differential equations which govern wave propagation and are used in modeling atmospheric and oceanic flows. The presentation establishes a concrete link between theory and practice.

Author: Krzysztof Murawski
Publisher: World Scientific
ISBN: 9814487562
Category : Science
Languages : en
Pages : 255

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Book Description
This book surveys analytical and numerical techniques appropriate to the description of fluid motion with an emphasis on the most widely used techniques exhibiting the best performance.Analytical and numerical solutions to hyperbolic systems of wave equations are the primary focus of the book. In addition, many interesting wave phenomena in fluids are considered using examples such as acoustic waves, the emission of air pollutants, magnetohydrodynamic waves in the solar corona, solar wind interaction with the planet venus, and ion-acoustic solitons.

Author: Randall J. LeVeque
Publisher: SIAM
ISBN: 9780898717839
Category : Mathematics
Languages : en
Pages : 356

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Book Description
This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.

Author: Gary Cohen
Publisher: Springer
ISBN: 9401777616
Category : Technology & Engineering
Languages : en
Pages : 393

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Book Description
This monograph presents numerical methods for solving transient wave equations (i.e. in time domain). More precisely, it provides an overview of continuous and discontinuous finite element methods for these equations, including their implementation in physical models, an extensive description of 2D and 3D elements with different shapes, such as prisms or pyramids, an analysis of the accuracy of the methods and the study of the Maxwell’s system and the important problem of its spurious free approximations. After recalling the classical models, i.e. acoustics, linear elastodynamics and electromagnetism and their variational formulations, the authors present a wide variety of finite elements of different shapes useful for the numerical resolution of wave equations. Then, they focus on the construction of efficient continuous and discontinuous Galerkin methods and study their accuracy by plane wave techniques and a priori error estimates. A chapter is devoted to the Maxwell’s system and the important problem of its spurious-free approximations. Treatment of unbounded domains by Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML) is described and analyzed in a separate chapter. The two last chapters deal with time approximation including local time-stepping and with the study of some complex models, i.e. acoustics in flow, gravity waves and vibrating thin plates. Throughout, emphasis is put on the accuracy and computational efficiency of the methods, with attention brought to their practical aspects.This monograph also covers in details the theoretical foundations and numerical analysis of these methods. As a result, this monograph will be of interest to practitioners, researchers, engineers and graduate students involved in the numerical simulationof waves.

Author: Dale R. Durran
Publisher: Springer Science & Business Media
ISBN: 1441964126
Category : Mathematics
Languages : en
Pages : 527

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Book Description
This scholarly text provides an introduction to the numerical methods used to model partial differential equations, with focus on atmospheric and oceanic flows. The book covers both the essentials of building a numerical model and the more sophisticated techniques that are now available. Finite difference methods, spectral methods, finite element method, flux-corrected methods and TVC schemes are all discussed. Throughout, the author keeps to a middle ground between the theorem-proof formalism of a mathematical text and the highly empirical approach found in some engineering publications. The book establishes a concrete link between theory and practice using an extensive range of test problems to illustrate the theoretically derived properties of various methods. From the reviews: "...the books unquestionable advantage is the clarity and simplicity in presenting virtually all basic ideas and methods of numerical analysis currently actively used in geophysical fluid dynamics." Physics of Atmosphere and Ocean

Author: Peter Moczo
Publisher: Cambridge University Press
ISBN: 1139867695
Category : Science
Languages : en
Pages : 387

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Book Description
Among all the numerical methods in seismology, the finite-difference (FD) technique provides the best balance of accuracy and computational efficiency. This book offers a comprehensive introduction to FD and its applications to earthquake motion. Using a systematic tutorial approach, the book requires only undergraduate degree-level mathematics and provides a user-friendly explanation of the relevant theory. It explains FD schemes for solving wave equations and elastodynamic equations of motion in heterogeneous media, and provides an introduction to the rheology of viscoelastic and elastoplastic media. It also presents an advanced FD time-domain method for efficient numerical simulations of earthquake ground motion in realistic complex models of local surface sedimentary structures. Accompanied by a suite of online resources to help put the theory into practice, this is a vital resource for professionals and academic researchers using numerical seismological techniques, and graduate students in earthquake seismology, computational and numerical modelling, and applied mathematics.