Analysis and Numerical Treatment of Highly Oscillatory Differential Equations PDF Download
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Languages : en
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Ce mémoire traite de la résolution (numérique et exacte) d'équations différentielles à grandes oscillations. On rencontre ce genre de problèmes en physique et en dynamique moléculaire, notamment. Ils sont modélisés par les équations hamiltoniennes p = - [delta][q souscrit]H(p, q), q =[delta][p souscrit]H(p, q). Ici H(p, q) est l'énergie totale et consiste en la somme des énergies (oscillatoires) d'oscillateurs harmoniques et d'un couplage. Les oscillateurs ont plusieurs groupes de fréquences: un groupe de petites fréquences et les autres de grandes fréquences. Résultats: (1) Etude de la presque conservation des énergies totale et oscillatoires pour la solution exacte du problème pour des temps longs à exponentiellement longs. (2) Développement de méthodes numériques adaptées aux problèmes hamiltoniens hautement oscillatoires. (3) Preuve de la presque conservation des énergies totale et oscillatoires pour la solution numérique.
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Category :
Languages : en
Pages :
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Book Description
Ce mémoire traite de la résolution (numérique et exacte) d'équations différentielles à grandes oscillations. On rencontre ce genre de problèmes en physique et en dynamique moléculaire, notamment. Ils sont modélisés par les équations hamiltoniennes p = - [delta][q souscrit]H(p, q), q =[delta][p souscrit]H(p, q). Ici H(p, q) est l'énergie totale et consiste en la somme des énergies (oscillatoires) d'oscillateurs harmoniques et d'un couplage. Les oscillateurs ont plusieurs groupes de fréquences: un groupe de petites fréquences et les autres de grandes fréquences. Résultats: (1) Etude de la presque conservation des énergies totale et oscillatoires pour la solution exacte du problème pour des temps longs à exponentiellement longs. (2) Développement de méthodes numériques adaptées aux problèmes hamiltoniens hautement oscillatoires. (3) Preuve de la presque conservation des énergies totale et oscillatoires pour la solution numérique.
Author: Xinyuan Wu
Publisher: Springer Nature
ISBN: 981160147X
Category : Mathematics
Languages : en
Pages : 507
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Book Description
The idea of structure-preserving algorithms appeared in the 1980's. The new paradigm brought many innovative changes. The new paradigm wanted to identify the long-time behaviour of the solutions or the existence of conservation laws or some other qualitative feature of the dynamics. Another area that has kept growing in importance within Geometric Numerical Integration is the study of highly-oscillatory problems: problems where the solutions are periodic or quasiperiodic and have to be studied in time intervals that include an extremely large number of periods. As is known, these equations cannot be solved efficiently using conventional methods. A further study of novel geometric integrators has become increasingly important in recent years. The objective of this monograph is to explore further geometric integrators for highly oscillatory problems that can be formulated as systems of ordinary and partial differential equations. Facing challenging scientific computational problems, this book presents some new perspectives of the subject matter based on theoretical derivations and mathematical analysis, and provides high-performance numerical simulations. In order to show the long-time numerical behaviour of the simulation, all the integrators presented in this monograph have been tested and verified on highly oscillatory systems from a wide range of applications in the field of science and engineering. They are more efficient than existing schemes in the literature for differential equations that have highly oscillatory solutions. This book is useful to researchers, teachers, students and engineers who are interested in Geometric Integrators and their long-time behaviour analysis for differential equations with highly oscillatory solutions.
Author: Marianna Khanamiryan
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Category :
Languages : en
Pages :
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This thesis presents methods for efficient numerical approximation of linear and non-linear systems of highly oscillatory ordinary differential equations. Phenomena of high oscillation is considered a major computational problem occurring in Fourier analysis, computational harmonic analysis, quantum mechanics, electrodynamics and fluid dynamics. Classical methods based on Gaussian quadrature fail to approximate oscillatory integrals. In this work we introduce numerical methods which share the remarkable feature that the accuracy of approximation improves as the frequency of oscillation increases. Asymptotically, our methods depend on inverse powers of the frequency of oscillation, turning the major computational problem into an advantage. Evolving ideas from the stationary phase method, we first apply the asymptotic method to solve highly oscillatory linear systems of differential equations. The asymptotic method provides a background for our next, the Filon-type method, which is highly accurate and requires computation of moments. We also introduce two novel methods. The first method, we call it the FM method, is a combination of Magnus approach and the Filon-type method, to solve matrix exponential. The second method, we call it the WRF method, a combination of the Filon-type method and the waveform relaxation methods, for solving highly oscillatory non-linear systems. Finally, completing the theory, we show that the Filon-type method can be replaced by a less accurate but moment free Levin-type method.
Author: Taketomo Mitsui
Publisher: World Scientific
ISBN: 9814500569
Category : Mathematics
Languages : en
Pages : 240
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Book Description
The book collects original articles on numerical analysis of ordinary differential equations and its applications. Some of the topics covered in this volume are: discrete variable methods, Runge-Kutta methods, linear multistep methods, stability analysis, parallel implementation, self-validating numerical methods, analysis of nonlinear oscillation by numerical means, differential-algebraic and delay-differential equations, and stochastic initial value problems.
Author: Simeon Ola Fatunla
Publisher: Academic Press
ISBN: 1483269264
Category : Mathematics
Languages : en
Pages : 308
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Book Description
Numerical Method for Initial Value Problems in Ordinary Differential Equations deals with numerical treatment of special differential equations: stiff, stiff oscillatory, singular, and discontinuous initial value problems, characterized by large Lipschitz constants. The book reviews the difference operators, the theory of interpolation, first integral mean value theorem, and numerical integration algorithms. The text explains the theory of one-step methods, the Euler scheme, the inverse Euler scheme, and also Richardson's extrapolation. The book discusses the general theory of Runge-Kutta processes, including the error estimation, and stepsize selection of the R-K process. The text evaluates the different linear multistep methods such as the explicit linear multistep methods (Adams-Bashforth, 1883), the implicit linear multistep methods (Adams-Moulton scheme, 1926), and the general theory of linear multistep methods. The book also reviews the existing stiff codes based on the implicit/semi-implicit, singly/diagonally implicit Runge-Kutta schemes, the backward differentiation formulas, the second derivative formulas, as well as the related extrapolation processes. The text is intended for undergraduates in mathematics, computer science, or engineering courses, andfor postgraduate students or researchers in related disciplines.
Author: Xinyuan Wu
Publisher: Springer
ISBN: 3662481561
Category : Technology & Engineering
Languages : en
Pages : 305
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Book Description
This book describes a variety of highly effective and efficient structure-preserving algorithms for second-order oscillatory differential equations. Such systems arise in many branches of science and engineering, and the examples in the book include systems from quantum physics, celestial mechanics and electronics. To accurately simulate the true behavior of such systems, a numerical algorithm must preserve as much as possible their key structural properties: time-reversibility, oscillation, symplecticity, and energy and momentum conservation. The book describes novel advances in RKN methods, ERKN methods, Filon-type asymptotic methods, AVF methods, and trigonometric Fourier collocation methods. The accuracy and efficiency of each of these algorithms are tested via careful numerical simulations, and their structure-preserving properties are rigorously established by theoretical analysis. The book also gives insights into the practical implementation of the methods. This book is intended for engineers and scientists investigating oscillatory systems, as well as for teachers and students who are interested in structure-preserving algorithms for differential equations.
Author: Xinyuan Wu
Publisher: Springer
ISBN: 9811090041
Category : Mathematics
Languages : en
Pages : 356
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Book Description
The main theme of this book is recent progress in structure-preserving algorithms for solving initial value problems of oscillatory differential equations arising in a variety of research areas, such as astronomy, theoretical physics, electronics, quantum mechanics and engineering. It systematically describes the latest advances in the development of structure-preserving integrators for oscillatory differential equations, such as structure-preserving exponential integrators, functionally fitted energy-preserving integrators, exponential Fourier collocation methods, trigonometric collocation methods, and symmetric and arbitrarily high-order time-stepping methods. Most of the material presented here is drawn from the recent literature. Theoretical analysis of the newly developed schemes shows their advantages in the context of structure preservation. All the new methods introduced in this book are proven to be highly effective compared with the well-known codes in the scientific literature. This book also addresses challenging problems at the forefront of modern numerical analysis and presents a wide range of modern tools and techniques.
Author: Sirakov Boyan
Publisher: World Scientific
ISBN: 9813272899
Category : Mathematics
Languages : en
Pages : 5396
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Book Description
The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.
Author: Bjorn Engquist
Publisher: Cambridge University Press
ISBN: 0521134439
Category : Mathematics
Languages : en
Pages : 254
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Book Description
Review papers from experts in areas of active research into highly oscillatory problems, with an emphasis on computation.
Author: Linda Ruth Petzold
Publisher:
ISBN:
Category : Eigenvalues
Languages : en
Pages : 288
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Book Description
