Spectral Methods For Time Dependent Problems

Author : Jan S. Hesthaven,Sigal Gottlieb,David Gottlieb
Publisher : Cambridge University Press
Page : 4 pages
File Size : 52,5 Mb
Release : 2007-01-11
Category : Mathematics
ISBN : 9781139459525

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Spectral Methods for Time-Dependent Problems by Jan S. Hesthaven,Sigal Gottlieb,David Gottlieb Pdf

Spectral methods are well-suited to solve problems modeled by time-dependent partial differential equations: they are fast, efficient and accurate and widely used by mathematicians and practitioners. This class-tested 2007 introduction, the first on the subject, is ideal for graduate courses, or self-study. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including discussions of stability, boundary conditions, filtering, and the extension from the linear to the nonlinear situation). Computational solution techniques for integration in time are dealt with by Runge-Kutta type methods. Several chapters are devoted to material not previously covered in book form, including stability theory for polynomial methods, techniques for problems with discontinuous solutions, round-off errors and the formulation of spectral methods on general grids. These will be especially helpful for practitioners.

Author : National Aeronautics and Space Administration (NASA)
Publisher : Createspace Independent Publishing Platform
Page : 74 pages
File Size : 46,9 Mb
Release : 2018-07-16
Category : Electronic
ISBN : 1722768150

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Spectral Methods for Time Dependent Problems by National Aeronautics and Space Administration (NASA) Pdf

Spectral approximations are reviewed for time dependent problems. Some basic ingredients from the spectral Fourier and Chebyshev approximations theory are discussed. A brief survey was made of hyperbolic and parabolic time dependent problems which are dealt with by both the energy method and the related Fourier analysis. The ideas presented above are combined in the study of accuracy stability and convergence of the spectral Fourier approximation to time dependent problems. Tadmor, Eitan Unspecified Center...

Author : Jan S. Hesthaven,Sigal Gottlieb,David Gottlieb
Publisher : Cambridge University Press
Page : 284 pages
File Size : 50,9 Mb
Release : 2007-01-11
Category : Mathematics
ISBN : 0521792118

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Spectral Methods for Time-Dependent Problems by Jan S. Hesthaven,Sigal Gottlieb,David Gottlieb Pdf

Spectral methods are well-suited to solve problems modeled by time-dependent partial differential equations: they are fast, efficient and accurate and widely used by mathematicians and practitioners. This class-tested 2007 introduction, the first on the subject, is ideal for graduate courses, or self-study. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including discussions of stability, boundary conditions, filtering, and the extension from the linear to the nonlinear situation). Computational solution techniques for integration in time are dealt with by Runge-Kutta type methods. Several chapters are devoted to material not previously covered in book form, including stability theory for polynomial methods, techniques for problems with discontinuous solutions, round-off errors and the formulation of spectral methods on general grids. These will be especially helpful for practitioners.

Author : John P. Boyd
Publisher : Courier Corporation
Page : 690 pages
File Size : 46,6 Mb
Release : 2001-12-03
Category : Mathematics
ISBN : 9780486411835

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Chebyshev and Fourier Spectral Methods by John P. Boyd Pdf

Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, methods for unbounded intervals, spherical and cylindrical geometry, and much more. 7 Appendices. Glossary. Bibliography. Index. Over 160 text figures.

Author : David A. Kopriva
Publisher : Springer Science & Business Media
Page : 397 pages
File Size : 40,8 Mb
Release : 2009-05-27
Category : Mathematics
ISBN : 9789048122615

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Implementing Spectral Methods for Partial Differential Equations by David A. Kopriva Pdf

This book explains how to solve partial differential equations numerically using single and multidomain spectral methods. It shows how only a few fundamental algorithms form the building blocks of any spectral code, even for problems with complex geometries.

Author : Bertil Gustafsson,Heinz-Otto Kreiss,Joseph Oliger
Publisher : John Wiley & Sons
Page : 464 pages
File Size : 44,6 Mb
Release : 2013-07-18
Category : Mathematics
ISBN : 9781118548523

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Time-Dependent Problems and Difference Methods by Bertil Gustafsson,Heinz-Otto Kreiss,Joseph Oliger Pdf

Praise for the First Edition ". . . fills a considerable gap in the numerical analysis literature by providing a self-contained treatment . . . this is an important work written in a clear style . . . warmly recommended to any graduate student or researcher in the field of the numerical solution of partial differential equations." —SIAM Review Time-Dependent Problems and Difference Methods, Second Edition continues to provide guidance for the analysis of difference methods for computing approximate solutions to partial differential equations for time-dependent problems. The book treats differential equations and difference methods with a parallel development, thus achieving a more useful analysis of numerical methods. The Second Edition presents hyperbolic equations in great detail as well as new coverage on second-order systems of wave equations including acoustic waves, elastic waves, and Einstein equations. Compared to first-order hyperbolic systems, initial-boundary value problems for such systems contain new properties that must be taken into account when analyzing stability. Featuring the latest material in partial differential equations with new theorems, examples, and illustrations,Time-Dependent Problems and Difference Methods, Second Edition also includes: High order methods on staggered grids Extended treatment of Summation By Parts operators and their application to second-order derivatives Simplified presentation of certain parts and proofs Time-Dependent Problems and Difference Methods, Second Edition is an ideal reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs and to predict and investigate physical phenomena. The book is also excellent for graduate-level courses in applied mathematics and scientific computations.

Author : Eitan Tadmor
Publisher : Unknown
Page : 76 pages
File Size : 41,7 Mb
Release : 1990
Category : Electronic
ISBN : NASA:31769000685795

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Spectral Methods for Time Dependent Problems by Eitan Tadmor Pdf

Author : Jie Shen,Tao Tang,Li-Lian Wang
Publisher : Springer Science & Business Media
Page : 481 pages
File Size : 51,6 Mb
Release : 2011-08-25
Category : Mathematics
ISBN : 9783540710417

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Spectral Methods by Jie Shen,Tao Tang,Li-Lian Wang Pdf

Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed presentation of basic spectral algorithms, as well as a systematical presentation of basic convergence theory and error analysis for spectral methods. Readers of this book will be exposed to a unified framework for designing and analyzing spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains. The book contains a large number of figures which are designed to illustrate various concepts stressed in the book. A set of basic matlab codes has been made available online to help the readers to develop their own spectral codes for their specific applications.

Author : Olivier Le Maitre,Omar M Knio
Publisher : Springer Science & Business Media
Page : 542 pages
File Size : 48,9 Mb
Release : 2010-03-11
Category : Science
ISBN : 9789048135202

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Spectral Methods for Uncertainty Quantification by Olivier Le Maitre,Omar M Knio Pdf

This book deals with the application of spectral methods to problems of uncertainty propagation and quanti?cation in model-based computations. It speci?cally focuses on computational and algorithmic features of these methods which are most useful in dealing with models based on partial differential equations, with special att- tion to models arising in simulations of ?uid ?ows. Implementations are illustrated through applications to elementary problems, as well as more elaborate examples selected from the authors’ interests in incompressible vortex-dominated ?ows and compressible ?ows at low Mach numbers. Spectral stochastic methods are probabilistic in nature, and are consequently rooted in the rich mathematical foundation associated with probability and measure spaces. Despite the authors’ fascination with this foundation, the discussion only - ludes to those theoretical aspects needed to set the stage for subsequent applications. The book is authored by practitioners, and is primarily intended for researchers or graduate students in computational mathematics, physics, or ?uid dynamics. The book assumes familiarity with elementary methods for the numerical solution of time-dependent, partial differential equations; prior experience with spectral me- ods is naturally helpful though not essential. Full appreciation of elaborate examples in computational ?uid dynamics (CFD) would require familiarity with key, and in some cases delicate, features of the associated numerical methods. Besides these shortcomings, our aim is to treat algorithmic and computational aspects of spectral stochastic methods with details suf?cient to address and reconstruct all but those highly elaborate examples.

Author : Bertil Gustafsson,Heinz-Otto Kreiss,Joseph Oliger
Publisher : John Wiley & Sons
Page : 666 pages
File Size : 53,5 Mb
Release : 1995
Category : Mathematics
ISBN : 0471507342

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Time Dependent Problems and Difference Methods by Bertil Gustafsson,Heinz-Otto Kreiss,Joseph Oliger Pdf

Time Dependent Problems and Difference Methods addresses these various industrial considerations in a pragmatic and detailed manner, giving special attention to time dependent problems in its coverage of the derivation and analysis of numerical methods for computational approximations to Partial Differential Equations (PDEs).

Author : David Gottlieb,Steven A. Orszag
Publisher : SIAM
Page : 167 pages
File Size : 52,7 Mb
Release : 1977-01-01
Category : Technology & Engineering
ISBN : 9780898710236

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Numerical Analysis of Spectral Methods by David Gottlieb,Steven A. Orszag Pdf

A unified discussion of the formulation and analysis of special methods of mixed initial boundary-value problems. The focus is on the development of a new mathematical theory that explains why and how well spectral methods work. Included are interesting extensions of the classical numerical analysis.

Author : Lloyd N. Trefethen
Publisher : SIAM
Page : 179 pages
File Size : 45,5 Mb
Release : 2000-07-01
Category : Mathematics
ISBN : 9780898714654

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Spectral Methods in MATLAB by Lloyd N. Trefethen Pdf

Mathematics of Computing -- Numerical Analysis.

Author : Hemen Dutta,Ahmet Ocak Akdemir,Abdon Atangana
Publisher : John Wiley & Sons
Page : 336 pages
File Size : 46,6 Mb
Release : 2020-08-06
Category : Mathematics
ISBN : 9781119654230

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Fractional Order Analysis by Hemen Dutta,Ahmet Ocak Akdemir,Abdon Atangana Pdf

A guide to the new research in the field of fractional order analysis Fractional Order Analysis contains the most recent research findings in fractional order analysis and its applications. The authors—noted experts on the topic—offer an examination of the theory, methods, applications, and the modern tools and techniques in the field of fractional order analysis. The information, tools, and applications presented can help develop mathematical methods and models with better accuracy. Comprehensive in scope, the book covers a range of topics including: new fractional operators, fractional derivatives, fractional differential equations, inequalities for different fractional derivatives and fractional integrals, fractional modeling related to transmission of Malaria, and dynamics of Zika virus with various fractional derivatives, and more. Designed to be an accessible text, several useful, relevant and connected topics can be found in one place, which is crucial for an understanding of the research problems of an applied nature. This book: Contains recent development in fractional calculus Offers a balance of theory, methods, and applications Puts the focus on fractional analysis and its interdisciplinary applications, such as fractional models for biological models Helps make research more relevant to real-life applications Written for researchers, professionals and practitioners, Fractional Order Analysis offers a comprehensive resource to fractional analysis and its many applications as well as information on the newest research.

Author : James F. Epperson
Publisher : John Wiley & Sons
Page : 676 pages
File Size : 46,9 Mb
Release : 2021-08-10
Category : Mathematics
ISBN : 9781119604693

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An Introduction to Numerical Methods and Analysis by James F. Epperson Pdf

The new edition of the popular introductory textbook on numerical approximation methods and mathematical analysis, with a unique emphasis on real-world application An Introduction to Numerical Methods and Analysis helps students gain a solid understanding of a wide range of numerical approximation methods for solving problems of mathematical analysis. Designed for entry-level courses on the subject, this popular textbook maximizes teaching flexibility by first covering basic topics before gradually moving to more advanced material in each chapter and section. Throughout the text, students are provided clear and accessible guidance on a wide range of numerical methods and analysis techniques, including root-finding, numerical integration, interpolation, solution of systems of equations, and many others. This fully revised third edition contains new sections on higher-order difference methods, the bisection and inertia method for computing eigenvalues of a symmetric matrix, a completely re-written section on different methods for Poisson equations, and spectral methods for higher-dimensional problems. New problem sets—ranging in difficulty from simple computations to challenging derivations and proofs—are complemented by computer programming exercises, illustrative examples, and sample code. This acclaimed textbook: Explains how to both construct and evaluate approximations for accuracy and performance Covers both elementary concepts and tools and higher-level methods and solutions Features new and updated material reflecting new trends and applications in the field Contains an introduction to key concepts, a calculus review, an updated primer on computer arithmetic, a brief history of scientific computing, a survey of computer languages and software, and a revised literature review Includes an appendix of proofs of selected theorems and a companion website with additional exercises, application models, and supplemental resources An Introduction to Numerical Methods and Analysis, Third Edition is the perfect textbook for upper-level undergraduate students in mathematics, science, and engineering courses, as well as for courses in the social sciences, medicine, and business with numerical methods and analysis components.

Author : George Rawitscher,Victo dos Santos Filho,Thiago Carvalho Peixoto
Publisher : Springer
Page : 221 pages
File Size : 49,5 Mb
Release : 2018-10-24
Category : Science
ISBN : 9783319427034

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An Introductory Guide to Computational Methods for the Solution of Physics Problems by George Rawitscher,Victo dos Santos Filho,Thiago Carvalho Peixoto Pdf

This monograph presents fundamental aspects of modern spectral and other computational methods, which are not generally taught in traditional courses. It emphasizes concepts as errors, convergence, stability, order and efficiency applied to the solution of physical problems. The spectral methods consist in expanding the function to be calculated into a set of appropriate basis functions (generally orthogonal polynomials) and the respective expansion coefficients are obtained via collocation equations. The main advantage of these methods is that they simultaneously take into account all available information, rather only the information available at a limited number of mesh points. They require more complicated matrix equations than those obtained in finite difference methods. However, the elegance, speed, and accuracy of the spectral methods more than compensates for any such drawbacks. During the course of the monograph, the authors examine the usually rapid convergence of the spectral expansions and the improved accuracy that results when nonequispaced support points are used, in contrast to the equispaced points used in finite difference methods. In particular, they demonstrate the enhanced accuracy obtained in the solutionof integral equations. The monograph includes an informative introduction to old and new computational methods with numerous practical examples, while at the same time pointing out the errors that each of the available algorithms introduces into the specific solution. It is a valuable resource for undergraduate students as an introduction to the field and for graduate students wishing to compare the available computational methods. In addition, the work develops the criteria required for students to select the most suitable method to solve the particular scientific problem that they are confronting.