Geometric Integrators For Differential Equations With Highly Oscillatory Solutions
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Author | : Xinyuan Wu |
Publisher | : Springer Nature |
Total Pages | : 507 |
Release | : 2021-09-28 |
Genre | : Mathematics |
ISBN | : 981160147X |
Download Geometric Integrators for Differential Equations with Highly Oscillatory Solutions Book in PDF, Epub and Kindle
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 | : Xinyuan Wu |
Publisher | : |
Total Pages | : 450 |
Release | : 2020 |
Genre | : Differential equations |
ISBN | : 9787030671127 |
Download Geometric Integrators for Differential Equations with Highly Oscillatory Solutions Book in PDF, Epub and Kindle
Author | : Ernst Hairer |
Publisher | : Springer Science & Business Media |
Total Pages | : 526 |
Release | : 2013-03-09 |
Genre | : Mathematics |
ISBN | : 3662050188 |
Download Geometric Numerical Integration Book in PDF, Epub and Kindle
This book deals with numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. 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 numerous figures, treats applications from physics and astronomy, and contains many numerical experiments and comparisons of different approaches.
Author | : Simeon Ola Fatunla |
Publisher | : |
Total Pages | : 36 |
Release | : 1977 |
Genre | : Differential equations |
ISBN | : |
Download Numerical Integrators for Stiff and Highly Oscillatory Differential Equations Book in PDF, Epub and Kindle
Author | : Xinyuan Wu |
Publisher | : Springer Science & Business Media |
Total Pages | : 244 |
Release | : 2013-02-02 |
Genre | : Technology & Engineering |
ISBN | : 364235338X |
Download Structure-Preserving Algorithms for Oscillatory Differential Equations Book in PDF, Epub and Kindle
Structure-Preserving Algorithms for Oscillatory Differential Equations describes a large number of highly effective and efficient structure-preserving algorithms for second-order oscillatory differential equations by using theoretical analysis and numerical validation. Structure-preserving algorithms for differential equations, especially for oscillatory differential equations, play an important role in the accurate simulation of oscillatory problems in applied sciences and engineering. The book discusses novel advances in the ARKN, ERKN, two-step ERKN, Falkner-type and energy-preserving methods, etc. for oscillatory differential equations. The work is intended for scientists, engineers, teachers and students who are interested in structure-preserving algorithms for differential equations. Xinyuan Wu is a professor at Nanjing University; Xiong You is an associate professor at Nanjing Agricultural University; Bin Wang is a joint Ph.D student of Nanjing University and University of Cambridge.
Author | : Benedict Leimkuhler |
Publisher | : Cambridge University Press |
Total Pages | : 464 |
Release | : 2004 |
Genre | : Mathematics |
ISBN | : 9780521772907 |
Download Simulating Hamiltonian Dynamics Book in PDF, Epub and Kindle
Geometric integrators are time-stepping methods, designed such that they exactly satisfy conservation laws, symmetries or symplectic properties of a system of differential equations. In this book the authors outline the principles of geometric integration and demonstrate how they can be applied to provide efficient numerical methods for simulating conservative models. Beginning from basic principles and continuing with discussions regarding the advantageous properties of such schemes, the book introduces methods for the N-body problem, systems with holonomic constraints, and rigid bodies. More advanced topics treated include high-order and variable stepsize methods, schemes for treating problems involving multiple time-scales, and applications to molecular dynamics and partial differential equations. The emphasis is on providing a unified theoretical framework as well as a practical guide for users. The inclusion of examples, background material and exercises enhance the usefulness of the book for self-instruction or as a text for a graduate course on the subject.
Author | : Wolf-Jürgen Beyn |
Publisher | : Springer |
Total Pages | : 324 |
Release | : 2013-12-12 |
Genre | : Mathematics |
ISBN | : 3319013009 |
Download Current Challenges in Stability Issues for Numerical Differential Equations Book in PDF, Epub and Kindle
This volume addresses some of the research areas in the general field of stability studies for differential equations, with emphasis on issues of concern for numerical studies. Topics considered include: (i) the long time integration of Hamiltonian Ordinary DEs and highly oscillatory systems, (ii) connection between stochastic DEs and geometric integration using the Markov chain Monte Carlo method, (iii) computation of dynamic patterns in evolutionary partial DEs, (iv) decomposition of matrices depending on parameters and localization of singularities, and (v) uniform stability analysis for time dependent linear initial value problems of ODEs. The problems considered in this volume are of interest to people working on numerical as well as qualitative aspects of differential equations, and it will serve both as a reference and as an entry point into further research.
Author | : Xinyuan Wu |
Publisher | : Springer |
Total Pages | : 356 |
Release | : 2018-04-19 |
Genre | : Mathematics |
ISBN | : 9811090041 |
Download Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations Book in PDF, Epub and Kindle
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 | : Xinyuan Wu |
Publisher | : Springer |
Total Pages | : 305 |
Release | : 2016-03-03 |
Genre | : Technology & Engineering |
ISBN | : 3662481561 |
Download Structure-Preserving Algorithms for Oscillatory Differential Equations II Book in PDF, Epub and Kindle
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 | : A. Iserles |
Publisher | : Cambridge University Press |
Total Pages | : 481 |
Release | : 2009 |
Genre | : Mathematics |
ISBN | : 0521734908 |
Download A First Course in the Numerical Analysis of Differential Equations Book in PDF, Epub and Kindle
lead the reader to a theoretical understanding of the subject without neglecting its practical aspects. The outcome is a textbook that is mathematically honest and rigorous and provides its target audience with a wide range of skills in both ordinary and partial differential equations." --Book Jacket.