Geometric Numerical Integration Of Differential Equations
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Geometric Numerical Integration
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Author : Ernst Hairer
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-05-18
Geometric Numerical Integration written by Ernst Hairer and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-05-18 with Mathematics categories.
This book covers numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. It presents a theory of symplectic and symmetric methods, which include various specially designed integrators, as well as discusses their construction and practical merits. The long-time behavior of the numerical solutions is studied using a backward error analysis combined with KAM theory.
A Concise Introduction To Geometric Numerical Integration
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Author : Sergio Blanes
language : en
Publisher: CRC Press
Release Date : 2017-11-22
A Concise Introduction To Geometric Numerical Integration written by Sergio Blanes and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-22 with Mathematics categories.
Discover How Geometric Integrators Preserve the Main Qualitative Properties of Continuous Dynamical Systems A Concise Introduction to Geometric Numerical Integration presents the main themes, techniques, and applications of geometric integrators for researchers in mathematics, physics, astronomy, and chemistry who are already familiar with numerical tools for solving differential equations. It also offers a bridge from traditional training in the numerical analysis of differential equations to understanding recent, advanced research literature on numerical geometric integration. The book first examines high-order classical integration methods from the structure preservation point of view. It then illustrates how to construct high-order integrators via the composition of basic low-order methods and analyzes the idea of splitting. It next reviews symplectic integrators constructed directly from the theory of generating functions as well as the important category of variational integrators. The authors also explain the relationship between the preservation of the geometric properties of a numerical method and the observed favorable error propagation in long-time integration. The book concludes with an analysis of the applicability of splitting and composition methods to certain classes of partial differential equations, such as the Schrödinger equation and other evolution equations. The motivation of geometric numerical integration is not only to develop numerical methods with improved qualitative behavior but also to provide more accurate long-time integration results than those obtained by general-purpose algorithms. Accessible to researchers and post-graduate students from diverse backgrounds, this introductory book gets readers up to speed on the ideas, methods, and applications of this field. Readers can reproduce the figures and results given in the text using the MATLAB® programs and model files available online.
Geometric Numerical Integration
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Author : Ernst Hairer
language : en
Publisher: Springer Verlag
Release Date : 2006-02-22
Geometric Numerical Integration written by Ernst Hairer and has been published by Springer Verlag this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-02-22 with Mathematics categories.
This book covers numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. It presents a theory of symplectic and symmetric methods, which include various specially designed integrators, as well as discusses their construction and practical merits. The long-time behavior of the numerical solutions is studied using a backward error analysis combined with KAM theory.
Geometric Numerical Integration
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Author : Ernst Hairer
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Geometric Numerical Integration written by Ernst Hairer and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.
Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches.
Geometric Numerical Integration Of Differential Equations
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Author : Reinout Quispel
language : en
Publisher:
Release Date : 2006
Geometric Numerical Integration Of Differential Equations written by Reinout Quispel and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with categories.
A Concise Introduction To Geometric Numerical Integration
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Author : Sergio Blanes
language : en
Publisher: CRC Press
Release Date : 2017-11-22
A Concise Introduction To Geometric Numerical Integration written by Sergio Blanes and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-22 with Mathematics categories.
Discover How Geometric Integrators Preserve the Main Qualitative Properties of Continuous Dynamical Systems A Concise Introduction to Geometric Numerical Integration presents the main themes, techniques, and applications of geometric integrators for researchers in mathematics, physics, astronomy, and chemistry who are already familiar with numerical tools for solving differential equations. It also offers a bridge from traditional training in the numerical analysis of differential equations to understanding recent, advanced research literature on numerical geometric integration. The book first examines high-order classical integration methods from the structure preservation point of view. It then illustrates how to construct high-order integrators via the composition of basic low-order methods and analyzes the idea of splitting. It next reviews symplectic integrators constructed directly from the theory of generating functions as well as the important category of variational integrators. The authors also explain the relationship between the preservation of the geometric properties of a numerical method and the observed favorable error propagation in long-time integration. The book concludes with an analysis of the applicability of splitting and composition methods to certain classes of partial differential equations, such as the Schrödinger equation and other evolution equations. The motivation of geometric numerical integration is not only to develop numerical methods with improved qualitative behavior but also to provide more accurate long-time integration results than those obtained by general-purpose algorithms. Accessible to researchers and post-graduate students from diverse backgrounds, this introductory book gets readers up to speed on the ideas, methods, and applications of this field. Readers can reproduce the figures and results given in the text using the MATLAB® programs and model files available online.
Special Issue On Geometric Numerical Integration Of Differential Equations
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Author :
language : en
Publisher:
Release Date : 2006
Special Issue On Geometric Numerical Integration Of Differential Equations written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Differential equations categories.
Geometric Integrators For Differential Equations With Highly Oscillatory Solutions
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Author : Xinyuan Wu
language : en
Publisher: Springer Nature
Release Date : 2021-09-28
Geometric Integrators For Differential Equations With Highly Oscillatory Solutions written by Xinyuan Wu and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-09-28 with Mathematics categories.
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.
Geometric Numerical Integration
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Author : Pui Sze Priscilla Tse
language : en
Publisher:
Release Date : 2007
Geometric Numerical Integration written by Pui Sze Priscilla Tse and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Differential equations categories.
This result implies that B-series methods cannot preserve both volume and affine symmetries simultaneously for arbitrary systems. Lastly, we present a new fourth-order method which is symplectic, self-adjoint, and preserves the exact linearization at fixed points of an arbitrary Hamiltonian system. This method is based on the preprocessed vector field integrators, which is the concept of integrating a preprocessed version of the original vector field as opposed to direct integration. We find numerically that, for certain Hamiltonian systems, the global truncation error grows more slowly for this new preprocessed integrator than for other fourth-order integrators such as the non-symplectic Runge-Kutta method and the symplectic and self-adjoint Gauss method.
Structure Preserving Algorithms For Oscillatory Differential Equations Ii
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Author : Xinyuan Wu
language : en
Publisher: Springer
Release Date : 2016-03-03
Structure Preserving Algorithms For Oscillatory Differential Equations Ii written by Xinyuan Wu and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-03-03 with Technology & Engineering categories.
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.