Almost Symplectic Runge-Kutta Schemes for Hamiltonian Systems PDF Download
Are you looking for read ebook online? Search for your book and save it on your Kindle device, PC, phones or tablets. Download Almost Symplectic Runge-Kutta Schemes for Hamiltonian Systems PDF full book. Access full book title Almost Symplectic Runge-Kutta Schemes for Hamiltonian Systems by . Download full books in PDF and EPUB format.
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 30
Get Book Here
Book Description
Symplectic Runge-Kutta schemes for the integration of general Hamiltonian systems are implicit. In practice one has to solve the implicit algebraic equations using some iterative approximation method, in which case the resulting integration scheme is no longer symplectic. In this paper we first analyze the preservation of the symplectic structure under two popular approximation schemes, fixed-point iteration and Newton's method, respectively. Error bounds for the symplectic structure are established when N fixed-point iterations or N iterations of Newton's method are used. The implications of these results for the implementation of symplectic methods are discussed and then explored through extensive numerical examples. Numerical comparisons with non-symplectic Runge-Kutta methods and pseudo-symplectic methods are also presented.
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 30
Get Book Here
Book Description
Symplectic Runge-Kutta schemes for the integration of general Hamiltonian systems are implicit. In practice one has to solve the implicit algebraic equations using some iterative approximation method, in which case the resulting integration scheme is no longer symplectic. In this paper we first analyze the preservation of the symplectic structure under two popular approximation schemes, fixed-point iteration and Newton's method, respectively. Error bounds for the symplectic structure are established when N fixed-point iterations or N iterations of Newton's method are used. The implications of these results for the implementation of symplectic methods are discussed and then explored through extensive numerical examples. Numerical comparisons with non-symplectic Runge-Kutta methods and pseudo-symplectic methods are also presented.
Author: Sergej B. Kuksin
Publisher: Springer
ISBN: 3540479201
Category : Mathematics
Languages : en
Pages : 128
Get Book Here
Book Description
The book is devoted to partial differential equations of Hamiltonian form, close to integrable equations. For such equations a KAM-like theorem is proved, stating that solutions of the unperturbed equation that are quasiperiodic in time mostly persist in the perturbed one. The theorem is applied to classical nonlinear PDE's with one-dimensional space variable such as the nonlinear string and nonlinear Schr|dinger equation andshow that the equations have "regular" (=time-quasiperiodic and time-periodic) solutions in rich supply. These results cannot be obtained by other techniques. The book will thus be of interest to mathematicians and physicists working with nonlinear PDE's. An extensivesummary of the results and of related topics is provided in the Introduction. All the nontraditional material used is discussed in the firstpart of the book and in five appendices.
Author: L. Jay
Publisher:
ISBN:
Category :
Languages : en
Pages : 30
Get Book Here
Book Description
Author: J.M. Sanz-Serna
Publisher: Courier Dover Publications
ISBN: 0486824101
Category : Mathematics
Languages : en
Pages : 225
Get Book Here
Book Description
Advanced text explores mathematical problems that occur frequently in physics and other sciences. Topics include symplectic integration, symplectic order conditions, available symplectic methods, numerical experiments, properties of symplectic integrators. 1994 edition.
Author: Laurent-Olivier Jay
Publisher:
ISBN:
Category :
Languages : en
Pages : 30
Get Book Here
Book Description
Author: Xinyuan Wu
Publisher: Springer Science & Business Media
ISBN: 364235338X
Category : Technology & Engineering
Languages : en
Pages : 244
Get Book Here
Book Description
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: Sebastian Reich
Publisher:
ISBN:
Category : Hamiltonian systems
Languages : en
Pages : 24
Get Book Here
Book Description
Again it turns out that those partitioned Runge-Kutta methods which are symplectic for unconstrained systems can be applied to constrained Hamiltonian systems. We show that, in contrast to implicit Runge-Kutta methods, the class of symplectic partitioned Runge-Kutta methods includes methods that also preserve the constraints. In the third part of the paper we discuss constrained Hamiltonian systems with separable Hamiltonian from a Lie algebraic point of view. This approach not only provides a different approach to the numerical integration of Hamiltonian systems but also allows for a straightforward backward error analysis."
Author: Xinyuan Wu
Publisher: Springer
ISBN: 3662481561
Category : Technology & Engineering
Languages : en
Pages : 305
Get Book Here
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: Carles Simó
Publisher: Springer Science & Business Media
ISBN: 9780792357100
Category : Mathematics
Languages : en
Pages : 690
Get Book Here
Book Description
A survey of current knowledge about Hamiltonian systems with three or more degrees of freedom and related topics. The Hamiltonian systems appearing in most of the applications are non-integrable. Hence methods to prove non-integrability results are presented and the different meaning attributed to non-integrability are discussed. For systems near an integrable one, it can be shown that, under suitable conditions, some parts of the integrable structure, most of the invariant tori, survive. Many of the papers discuss near-integrable systems. From a topological point of view, some singularities must appear in different problems, either caustics, geodesics, moving wavefronts, etc. This is also related to singularities in the projections of invariant objects, and can be used as a signature of these objects. Hyperbolic dynamics appear as a source on unpredictable behaviour and several mechanisms of hyperbolicity are presented. The destruction of tori leads to Aubrey-Mather objects, and this is touched on for a related class of systems. Examples without periodic orbits are constructed, against a classical conjecture. Other topics concern higher dimensional systems, either finite (networks and localised vibrations on them) or infinite, like the quasiperiodic Schrödinger operator or nonlinear hyperbolic PDE displaying quasiperiodic solutions. Most of the applications presented concern celestial mechanics problems, like the asteroid problem, the design of spacecraft orbits, and methods to compute periodic solutions.
Author: Mich'le Audin
Publisher: American Mathematical Soc.
ISBN: 9780821844137
Category : Mathematics
Languages : en
Pages : 172
Get Book Here
Book Description
"This book presents some modern techniques in the theory of integrable systems viewed as variations on the theme of action-angle coordinates. These techniques include analytical methods coming from the Galois theory of differential equations, as well as more classical algebro-geometric methods related to Lax equations. This book would be suitable for a graduate course in Hamiltonian systems."--BOOK JACKET.
