Geometric Numerical Integration

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Geometric Numerical Integration

Ernst Hairer,Christian Lubich,Gerhard Wanner
Geometric Numerical Integration Book PDF

Author : Ernst Hairer,Christian Lubich,Gerhard Wanner
Publisher : Springer Science & Business Media
Page : 526 pages
File Size : 54,8 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662050187

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Geometric Numerical Integration by Ernst Hairer,Christian Lubich,Gerhard Wanner Pdf

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.

A Concise Introduction to Geometric Numerical Integration

Sergio Blanes,Fernando Casas
A Concise Introduction to Geometric Numerical Integration Book PDF

Author : Sergio Blanes,Fernando Casas
Publisher : CRC Press
Page : 218 pages
File Size : 44,9 Mb
Release : 2017-11-22
Category : Mathematics
ISBN : 9781315354866

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A Concise Introduction to Geometric Numerical Integration by Sergio Blanes,Fernando Casas Pdf

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.

A Concise Introduction to Geometric Numerical Integration

Sergio Blanes,Fernando Casas
A Concise Introduction to Geometric Numerical Integration Book PDF

Author : Sergio Blanes,Fernando Casas
Publisher : CRC Press
Page : 233 pages
File Size : 50,7 Mb
Release : 2017-11-22
Category : Mathematics
ISBN : 9781482263442

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A Concise Introduction to Geometric Numerical Integration by Sergio Blanes,Fernando Casas Pdf

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.

Symplectic Geometric Algorithms for Hamiltonian Systems

Kang Feng,Mengzhao Qin
Symplectic Geometric Algorithms for Hamiltonian Systems Book PDF

Author : Kang Feng,Mengzhao Qin
Publisher : Springer Science & Business Media
Page : 690 pages
File Size : 49,8 Mb
Release : 2010-10-18
Category : Mathematics
ISBN : 9783642017773

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Symplectic Geometric Algorithms for Hamiltonian Systems by Kang Feng,Mengzhao Qin Pdf

"Symplectic Geometric Algorithms for Hamiltonian Systems" will be useful not only for numerical analysts, but also for those in theoretical physics, computational chemistry, celestial mechanics, etc. The book generalizes and develops the generating function and Hamilton-Jacobi equation theory from the perspective of the symplectic geometry and symplectic algebra. It will be a useful resource for engineers and scientists in the fields of quantum theory, astrophysics, atomic and molecular dynamics, climate prediction, oil exploration, etc. Therefore a systematic research and development of numerical methodology for Hamiltonian systems is well motivated. Were it successful, it would imply wide-ranging applications.

Numerical Geometry of Images

Ron Kimmel
Numerical Geometry of Images Book PDF

Author : Ron Kimmel
Publisher : Springer Science & Business Media
Page : 222 pages
File Size : 47,8 Mb
Release : 2012-09-07
Category : Computers
ISBN : 9780387216379

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Numerical Geometry of Images by Ron Kimmel Pdf

Numerical Geometry of Images examines computational methods and algorithms in image processing. It explores applications like shape from shading, color-image enhancement and segmentation, edge integration, offset curve computation, symmetry axis computation, path planning, minimal geodesic computation, and invariant signature calculation. In addition, it describes and utilizes tools from mathematical morphology, differential geometry, numerical analysis, and calculus of variations. Graduate students, professionals, and researchers with interests in computational geometry, image processing, computer graphics, and algorithms will find this new text / reference an indispensable source of insight of instruction.

Simulating Hamiltonian Dynamics

Benedict Leimkuhler,Sebastian Reich
Simulating Hamiltonian Dynamics Book PDF

Author : Benedict Leimkuhler,Sebastian Reich
Publisher : Cambridge University Press
Page : 464 pages
File Size : 53,7 Mb
Release : 2004
Category : Mathematics
ISBN : 0521772907

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Simulating Hamiltonian Dynamics by Benedict Leimkuhler,Sebastian Reich Pdf

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.

Geometric Numerical Integration and Schrödinger Equations

Erwan Faou
Geometric Numerical Integration and Schr dinger Equations Book PDF

Author : Erwan Faou
Publisher : European Mathematical Society
Page : 152 pages
File Size : 40,9 Mb
Release : 2012
Category : Numerical integration
ISBN : 3037191007

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Geometric Numerical Integration and Schrödinger Equations by Erwan Faou Pdf

The goal of geometric numerical integration is the simulation of evolution equations possessing geometric properties over long periods of time. Of particular importance are Hamiltonian partial differential equations typically arising in application fields such as quantum mechanics or wave propagation phenomena. They exhibit many important dynamical features such as energy preservation and conservation of adiabatic invariants over long periods of time. In this setting, a natural question is how and to which extent the reproduction of such long-time qualitative behavior can be ensured by numerical schemes. Starting from numerical examples, these notes provide a detailed analysis of the Schrodinger equation in a simple setting (periodic boundary conditions, polynomial nonlinearities) approximated by symplectic splitting methods. Analysis of stability and instability phenomena induced by space and time discretization are given, and rigorous mathematical explanations are provided for them. The book grew out of a graduate-level course and is of interest to researchers and students seeking an introduction to the subject matter.

A First Course in the Numerical Analysis of Differential Equations

Arieh Iserles
A First Course in the Numerical Analysis of Differential Equations Book PDF

Author : Arieh Iserles
Publisher : Cambridge University Press
Page : 481 pages
File Size : 47,5 Mb
Release : 2008-11-27
Category : Mathematics
ISBN : 9781139473767

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A First Course in the Numerical Analysis of Differential Equations by Arieh Iserles Pdf

Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This second edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems.

Geometric Integration Theory

Steven G. Krantz,Harold R. Parks
Geometric Integration Theory Book PDF

Author : Steven G. Krantz,Harold R. Parks
Publisher : Springer Science & Business Media
Page : 340 pages
File Size : 41,9 Mb
Release : 2008-12-15
Category : Mathematics
ISBN : 9780817646790

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Geometric Integration Theory by Steven G. Krantz,Harold R. Parks Pdf

This textbook introduces geometric measure theory through the notion of currents. Currents, continuous linear functionals on spaces of differential forms, are a natural language in which to formulate types of extremal problems arising in geometry, and can be used to study generalized versions of the Plateau problem and related questions in geometric analysis. Motivating key ideas with examples and figures, this book is a comprehensive introduction ideal for both self-study and for use in the classroom. The exposition demands minimal background, is self-contained and accessible, and thus is ideal for both graduate students and researchers.

Foundations of Computational Mathematics

Ronald A. DeVore,Arieh Iserles,Endre Süli
Foundations of Computational Mathematics Book PDF

Author : Ronald A. DeVore,Arieh Iserles,Endre Süli
Publisher : Cambridge University Press
Page : 418 pages
File Size : 48,9 Mb
Release : 2001-05-17
Category : Mathematics
ISBN : 0521003490

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Foundations of Computational Mathematics by Ronald A. DeVore,Arieh Iserles,Endre Süli Pdf

Collection of papers by leading researchers in computational mathematics, suitable for graduate students and researchers.

The Riemann Approach to Integration

Washek F. Pfeffer
The Riemann Approach to Integration Book PDF

Author : Washek F. Pfeffer
Publisher : Cambridge University Press
Page : 326 pages
File Size : 45,8 Mb
Release : 1993
Category : Mathematics
ISBN : 0521440351

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The Riemann Approach to Integration by Washek F. Pfeffer Pdf

A detailed exposition of generalised Riemann-Stieltjes integrals.

Line Integral Methods for Conservative Problems

Luigi Brugnano,Felice Iavernaro
Line Integral Methods for Conservative Problems Book PDF

Author : Luigi Brugnano,Felice Iavernaro
Publisher : CRC Press
Page : 222 pages
File Size : 44,5 Mb
Release : 2016-03-09
Category : Mathematics
ISBN : 9781482263855

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Line Integral Methods for Conservative Problems by Luigi Brugnano,Felice Iavernaro Pdf

Line Integral Methods for Conservative Problems explains the numerical solution of differential equations within the framework of geometric integration, a branch of numerical analysis that devises numerical methods able to reproduce (in the discrete solution) relevant geometric properties of the continuous vector field. The book focuses on a large set of differential systems named conservative problems, particularly Hamiltonian systems. Assuming only basic knowledge of numerical quadrature and Runge–Kutta methods, this self-contained book begins with an introduction to the line integral methods. It describes numerous Hamiltonian problems encountered in a variety of applications and presents theoretical results concerning the main instance of line integral methods: the energy-conserving Runge–Kutta methods, also known as Hamiltonian boundary value methods (HBVMs). The authors go on to address the implementation of HBVMs in order to recover in the numerical solution what was expected from the theory. The book also covers the application of HBVMs to handle the numerical solution of Hamiltonian partial differential equations (PDEs) and explores extensions of the energy-conserving methods. With many examples of applications, this book provides an accessible guide to the subject yet gives you enough details to allow concrete use of the methods. MATLAB codes for implementing the methods are available online.

Discrete Mechanics, Geometric Integration and Lie–Butcher Series

Kurusch Ebrahimi-Fard,María Barbero Liñán
Discrete Mechanics Geometric Integration and Lie Butcher Series Book PDF

Author : Kurusch Ebrahimi-Fard,María Barbero Liñán
Publisher : Springer
Page : 361 pages
File Size : 50,9 Mb
Release : 2018-11-05
Category : Mathematics
ISBN : 9783030013974

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Discrete Mechanics, Geometric Integration and Lie–Butcher Series by Kurusch Ebrahimi-Fard,María Barbero Liñán Pdf

This volume resulted from presentations given at the international “Brainstorming Workshop on New Developments in Discrete Mechanics, Geometric Integration and Lie–Butcher Series”, that took place at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid, Spain. It combines overview and research articles on recent and ongoing developments, as well as new research directions. Why geometric numerical integration? In their article of the same title Arieh Iserles and Reinout Quispel, two renowned experts in numerical analysis of differential equations, provide a compelling answer to this question. After this introductory chapter a collection of high-quality research articles aim at exploring recent and ongoing developments, as well as new research directions in the areas of geometric integration methods for differential equations, nonlinear systems interconnections, and discrete mechanics. One of the highlights is the unfolding of modern algebraic and combinatorial structures common to those topics, which give rise to fruitful interactions between theoretical as well as applied and computational perspectives. The volume is aimed at researchers and graduate students interested in theoretical and computational problems in geometric integration theory, nonlinear control theory, and discrete mechanics.

Geometric Discrepancy

Jiri Matousek
Geometric Discrepancy Book PDF

Author : Jiri Matousek
Publisher : Springer Science & Business Media
Page : 293 pages
File Size : 48,7 Mb
Release : 2009-12-02
Category : Mathematics
ISBN : 9783642039423

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Geometric Discrepancy by Jiri Matousek Pdf

What is the "most uniform" way of distributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? This book is an accessible and lively introduction to the area of geometric discrepancy theory, with numerous exercises and illustrations. In separate, more specialized parts, it also provides a comprehensive guide to recent research.

Geometric Integrators for Differential Equations with Highly Oscillatory Solutions

Xinyuan Wu,Bin Wang
Geometric Integrators for Differential Equations with Highly Oscillatory Solutions Book PDF

Author : Xinyuan Wu,Bin Wang
Publisher : Springer Nature
Page : 507 pages
File Size : 46,6 Mb
Release : 2021-09-28
Category : Mathematics
ISBN : 9789811601477

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Geometric Integrators for Differential Equations with Highly Oscillatory Solutions by Xinyuan Wu,Bin Wang Pdf

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.