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Author: L. N. Trefethen Publisher: ISBN: Category : Boundary value problems Languages : en Pages : 224
Book Description
This dissertation investigates the behavior of finite difference models of linear hyperbolic partial differential equations. Whereas a hyperbolic equation is nondispersive and nondissipative, difference models are invariably dispersive, and often dissipative too. We set about analyzing them by means of existing techniques from the theory of dispersive wave propagation, making extensive use in particular of the concept of group velocity, the velocity at which energy propagates. The first three chapters present a general analysis of wave propagation in difference models. We describe systematically the effects of dispersion on numerical errors, for both smooth and parasitic waves. The reflection and transmission of waves at boundaries and interfaces are then studied at length. The key point for this is a distinction introduced here between leftgoing and rightgoing signals, which is based not on the characteristics of the original equation, but on the group velocities of the numerical model. The last three chapters examine stability for finite difference models of initial boundary value problems.
Author: L. N. Trefethen Publisher: ISBN: Category : Boundary value problems Languages : en Pages : 224
Book Description
This dissertation investigates the behavior of finite difference models of linear hyperbolic partial differential equations. Whereas a hyperbolic equation is nondispersive and nondissipative, difference models are invariably dispersive, and often dissipative too. We set about analyzing them by means of existing techniques from the theory of dispersive wave propagation, making extensive use in particular of the concept of group velocity, the velocity at which energy propagates. The first three chapters present a general analysis of wave propagation in difference models. We describe systematically the effects of dispersion on numerical errors, for both smooth and parasitic waves. The reflection and transmission of waves at boundaries and interfaces are then studied at length. The key point for this is a distinction introduced here between leftgoing and rightgoing signals, which is based not on the characteristics of the original equation, but on the group velocities of the numerical model. The last three chapters examine stability for finite difference models of initial boundary value problems.
Author: D. Lee Publisher: Elsevier ISBN: 1483295699 Category : Science Languages : en Pages : 134
Book Description
A concise guide to the theory and application of numerical methods for predicting ocean acoustic propagation, also providing an introduction to numerical methods, with an overview of those methods presently in use. An in-depth development of the implicit-finite-difference technique is presented together with bench-mark test examples included to demonstrate its application to realistic ocean environments. Other applications include atmospheric acoustics, plasma physics, quantum mechanics, optics and seismology.
Author: Gary Cohen Publisher: Springer Science & Business Media ISBN: 366204823X Category : Science Languages : en Pages : 355
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: K. Murawski Publisher: World Scientific ISBN: 9789812776631 Category : Science Languages : en Pages : 260
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: Krzysztof Murawski Publisher: World Scientific ISBN: 9814487562 Category : Science Languages : en Pages : 255
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: Peter Moczo Publisher: Cambridge University Press ISBN: 1139867695 Category : Science Languages : en Pages : 387
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.
Author: Nikolaos A. Kampanis Publisher: CRC Press ISBN: 1420010875 Category : Mathematics Languages : en Pages : 707
Book Description
Due to the increase in computational power and new discoveries in propagation phenomena for linear and nonlinear waves, the area of computational wave propagation has become more significant in recent years. Exploring the latest developments in the field, Effective Computational Methods for Wave Propagation presents several modern, valuable
Author: Henry Martin Jurgens Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
The numerical simulation of linear wave propagation and scattering cannot be handled effectively using the second-order methods that are in widespread use today. To obtain accurate results these methods require grids with a resolution in the hundreds of points per wavelength to propagate a wave a distance greater than thirty wavelengths. Methods which can obtain accurate solutions without such high grid resolutions are needed for simulating long-range wave propagation. A procedure is presented for developing high-accuracy finite-difference schemes which are optimized in order to minimize the numerical phase and amplitude errors over a given range of frequencies. These high-accuracy schemes are used to solve the two-dimensional time-domain Maxwell equations for electromagnetic wave propagation and scattering. Boundary conditions are presented which preserve the accuracy of these schemes when modeling interfaces between different materials. A stable and accurate far-field boundary treatment is used which allows waves to exit the numerical domain with very little spurious reflection. Numerical experiments are performed which demonstrate the utility of the high-accuracy schemes using Cartesian and curvilinear grids. Two schemes are studied, one which produces the maximum formal order of accuracy, and one which is optimized for propagation distances less than roughly three hundred wavelengths. The high-accuracy schemes are shown to be substantially more efficient, in both computing time and memory, than methods which are second- and fourth-order in space. The optimized scheme can produce significant error reduction relative to the maximum-order scheme, with no additional expense, especially when the number of wavelengths of travel is large.
Author: Yajun An Publisher: ISBN: Category : Languages : en Pages : 96
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: L. N. Trefethen
Author: L. N. Trefethen
Author: D. Lee
Author: Gary Cohen
Author: K. Murawski
Author: Krzysztof Murawski
Author: Peter Moczo
Author: Nikolaos A. Kampanis
Author: Henry Martin Jurgens
Author: John C. Strikwerda
Author: Yajun An