Quasi Spectral Finite Difference Methods PDF Download
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Author | : Tristan Kremp |
Publisher | : Cuvillier Verlag |
Total Pages | : 240 |
Release | : 2011-03-25 |
Genre | : Mathematics |
ISBN | : 3736936850 |
Download Quasi-spectral Finite Difference Methods Book in PDF, ePub and Kindle
The doctoral thesis „Quasi-spectral finite difference methods: Convergence analysis and application to nonlinear optical pulse propagation“ by Tristan Kremp addresses the theory and application of so-called quasi-spectral finite differences. Contrary to the common Taylor approach, these are by construction exact for trigonometric instead of algebraic polynomials. With any fixed discretization spacing, this allows for a higher accuracy, e.g., when differencing functions that have a band-pass like Fourier spectrum. In this dissertation, the convergence of such quasi-spectral finite differences is proven for the first time. It is shown that the highest possible order of convergence is the same as for the Taylor approach, i.e., it is basically identical to the total number of summands in the finite difference. This order is achieved if all frequencies, for which the quasi-spectral finite difference is exact, vanish sufficiently fast in comparison to the discretization spacing. This condition can be easily incorporated in the finite difference weights construction, which can be achieved by spectral interpolation or least-squares optimization, respectively. Employing previously unknown Haar (or Chebyshev) systems that consist of combinations of algebraic and trigonometric monomials, the equivalence of both methods of construction is proven. In a semidiscretization framework, these finite differences are, for the first time, combined with exponential split-step integrators for an efficient solution of linear or nonlinear evolution equations. It is shown that a simple modification of the common symmetric split-step integrator guarantees its second-order convergence even in the presence of general nonlinearities. An important example of such a partial differential equation is the nonlinear Schrödinger equation (NLSE). In contrast to the standard literature, the NLSE is derived here directly from Maxwell’s equations, without the common assumption that the second spatial derivative in the direction of the propagation can be neglected, and without the assumption that the multiplicative nonlinear term behaves as a constant with respect to the Fourier transformation. A practically relevant application is the propagation of wavelength division multiplexing (WDM) signals in optical fibers. Compared to other semidiscretization techniques such as finite elements, wavelet collocation and the pseudo-spectral methods (split-step Fourier method) that are mostly employed by the industry, the quasi-spectral finite differences allow, at the same accuracy, for a substantial reduction of the computation time.
Author | : John P. Boyd |
Publisher | : Courier Corporation |
Total Pages | : 690 |
Release | : 2001-12-03 |
Genre | : Mathematics |
ISBN | : 0486411834 |
Download Chebyshev and Fourier Spectral Methods Book in PDF, ePub and Kindle
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 | : M. Y. Hussaini |
Publisher | : |
Total Pages | : 68 |
Release | : 1986 |
Genre | : Viscous flow |
ISBN | : |
Download Spectral Methods in Fluid Dynamics Book in PDF, ePub and Kindle
Author | : Jie Shen |
Publisher | : Springer Science & Business Media |
Total Pages | : 481 |
Release | : 2011-08-25 |
Genre | : Mathematics |
ISBN | : 3540710418 |
Download Spectral Methods Book in PDF, ePub and Kindle
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 | : Lloyd N. Trefethen |
Publisher | : SIAM |
Total Pages | : 179 |
Release | : 2000-07-01 |
Genre | : Mathematics |
ISBN | : 0898714656 |
Download Spectral Methods in MATLAB Book in PDF, ePub and Kindle
Mathematics of Computing -- Numerical Analysis.
Author | : Bengt Fornberg |
Publisher | : Cambridge University Press |
Total Pages | : 248 |
Release | : 1998-10-28 |
Genre | : Mathematics |
ISBN | : 9780521645645 |
Download A Practical Guide to Pseudospectral Methods Book in PDF, ePub and Kindle
This book explains how, when and why the pseudospectral approach works.
Author | : Tristan Kremp |
Publisher | : |
Total Pages | : 230 |
Release | : 2011 |
Genre | : |
ISBN | : 9783869556857 |
Download Quasi-spectral Finite Difference Methods Book in PDF, ePub and Kindle
Author | : Jie Shen |
Publisher | : |
Total Pages | : 224 |
Release | : 2006 |
Genre | : Calculus |
ISBN | : 9787030177223 |
Download Spectral and High-order Methods with Applications Book in PDF, ePub and Kindle
中国科学院科学出版基金资助出版。
Author | : Zhilin Li |
Publisher | : Cambridge University Press |
Total Pages | : 305 |
Release | : 2017-11-30 |
Genre | : Mathematics |
ISBN | : 1107163226 |
Download Numerical Solution of Differential Equations Book in PDF, ePub and Kindle
A practical and concise guide to finite difference and finite element methods. Well-tested MATLAB® codes are available online.
Author | : G. Evans |
Publisher | : Springer Science & Business Media |
Total Pages | : 308 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1447103793 |
Download Analytic Methods for Partial Differential Equations Book in PDF, ePub and Kindle
This is the practical introduction to the analytical approach taken in Volume 2. Based upon courses in partial differential equations over the last two decades, the text covers the classic canonical equations, with the method of separation of variables introduced at an early stage. The characteristic method for first order equations acts as an introduction to the classification of second order quasi-linear problems by characteristics. Attention then moves to different co-ordinate systems, primarily those with cylindrical or spherical symmetry. Hence a discussion of special functions arises quite naturally, and in each case the major properties are derived. The next section deals with the use of integral transforms and extensive methods for inverting them, and concludes with links to the use of Fourier series.