A Concise Introduction To Geometric Numerical Integration

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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 : 46,9 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.

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 : 660 pages
File Size : 46,9 Mb
Release : 2006-05-18
Category : Mathematics
ISBN : 9783540306665

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

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 Numerical Analysis

A. C. Faul
A Concise Introduction to Numerical Analysis Book PDF

Author : A. C. Faul
Publisher : CRC Press
Page : 161 pages
File Size : 44,7 Mb
Release : 2018-10-24
Category : Mathematics
ISBN : 9781498712217

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A Concise Introduction to Numerical Analysis by A. C. Faul Pdf

This textbook provides an accessible and concise introduction to numerical analysis for upper undergraduate and beginning graduate students from various backgrounds. It was developed from the lecture notes of four successful courses on numerical analysis taught within the MPhil of Scientific Computing at the University of Cambridge. The book is easily accessible, even to those with limited knowledge of mathematics. Students will get a concise, but thorough introduction to numerical analysis. In addition the algorithmic principles are emphasized to encourage a deeper understanding of why an algorithm is suitable, and sometimes unsuitable, for a particular problem. A Concise Introduction to Numerical Analysis strikes a balance between being mathematically comprehensive, but not overwhelming with mathematical detail. In some places where further detail was felt to be out of scope of the book, the reader is referred to further reading. The book uses MATLAB® implementations to demonstrate the workings of the method and thus MATLAB's own implementations are avoided, unless they are used as building blocks of an algorithm. In some cases the listings are printed in the book, but all are available online on the book’s page at www.crcpress.com. Most implementations are in the form of functions returning the outcome of the algorithm. Also, examples for the use of the functions are given. Exercises are included in line with the text where appropriate, and each chapter ends with a selection of revision exercises. Solutions to odd-numbered exercises are also provided on the book’s page at www.crcpress.com. This textbook is also an ideal resource for graduate students coming from other subjects who will use numerical techniques extensively in their graduate studies.

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 : 46,8 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.

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 : 55,7 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.

Numerical Approximation of Ordinary Differential Problems

Raffaele D'Ambrosio
Numerical Approximation of Ordinary Differential Problems Book PDF

Author : Raffaele D'Ambrosio
Publisher : Springer Nature
Page : 391 pages
File Size : 54,8 Mb
Release : 2023-09-26
Category : Mathematics
ISBN : 9783031313431

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Numerical Approximation of Ordinary Differential Problems by Raffaele D'Ambrosio Pdf

This book is focused on the numerical discretization of ordinary differential equations (ODEs), under several perspectives. The attention is first conveyed to providing accurate numerical solutions of deterministic problems. Then, the presentation moves to a more modern vision of numerical approximation, oriented to reproducing qualitative properties of the continuous problem along the discretized dynamics over long times. The book finally performs some steps in the direction of stochastic differential equations (SDEs), with the intention of offering useful tools to generalize the techniques introduced for the numerical approximation of ODEs to the stochastic case, as well as of presenting numerical issues natively introduced for SDEs. The book is the result of an intense teaching experience as well as of the research carried out in the last decade by the author. It is both intended for students and instructors: for the students, this book is comprehensive and rather self-contained; for the instructors, there is material for one or more monographic courses on ODEs and related topics. In this respect, the book can be followed in its designed path and includes motivational aspects, historical background, examples and a software programs, implemented in Matlab, that can be useful for the laboratory part of a course on numerical ODEs/SDEs. The book also contains the portraits of several pioneers in the numerical discretization of differential problems, useful to provide a framework to understand their contributes in the presented fields. Last, but not least, rigor joins readability in the book.

Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science

Roderick Melnik,Roman Makarov,Jacques Belair
Recent Progress and Modern Challenges in Applied Mathematics Modeling and Computational Science Book PDF

Author : Roderick Melnik,Roman Makarov,Jacques Belair
Publisher : Springer
Page : 444 pages
File Size : 54,8 Mb
Release : 2017-09-05
Category : Mathematics
ISBN : 9781493969692

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Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science by Roderick Melnik,Roman Makarov,Jacques Belair Pdf

This volume is an excellent resource for professionals in various areas of applications of mathematics, modeling, and computational science. It focuses on recent progress and modern challenges in these areas. The volume provides a balance between fundamental theoretical and applied developments, emphasizing the interdisciplinary nature of modern trends and detailing state-of-the-art achievements in Applied Mathematics, Modeling, and Computational Science. The chapters have been authored by international experts in their respective fields, making this book ideal for researchers in academia, practitioners, and graduate students. It can also serve as a reference in the diverse selected areas of applied mathematics, modelling, and computational sciences, and is ideal for interdisciplinary collaborations.

Introduction to Abelian Model Structures and Gorenstein Homological Dimensions

Marco A. P. Bullones
Introduction to Abelian Model Structures and Gorenstein Homological Dimensions Book PDF

Author : Marco A. P. Bullones
Publisher : CRC Press
Page : 370 pages
File Size : 42,6 Mb
Release : 2016-08-19
Category : Mathematics
ISBN : 9781498725354

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Introduction to Abelian Model Structures and Gorenstein Homological Dimensions by Marco A. P. Bullones Pdf

Introduction to Abelian Model Structures and Gorenstein Homological Dimensions provides a starting point to study the relationship between homological and homotopical algebra, a very active branch of mathematics. The book shows how to obtain new model structures in homological algebra by constructing a pair of compatible complete cotorsion pairs related to a specific homological dimension and then applying the Hovey Correspondence to generate an abelian model structure. The first part of the book introduces the definitions and notations of the universal constructions most often used in category theory. The next part presents a proof of the Eklof and Trlifaj theorem in Grothedieck categories and covers M. Hovey’s work that connects the theories of cotorsion pairs and model categories. The final two parts study the relationship between model structures and classical and Gorenstein homological dimensions and explore special types of Grothendieck categories known as Gorenstein categories. As self-contained as possible, this book presents new results in relative homological algebra and model category theory. The author also re-proves some established results using different arguments or from a pedagogical point of view. In addition, he proves folklore results that are difficult to locate in the literature.

Analytical Methods for Kolmogorov Equations

Luca Lorenzi
Analytical Methods for Kolmogorov Equations Book PDF

Author : Luca Lorenzi
Publisher : CRC Press
Page : 572 pages
File Size : 43,5 Mb
Release : 2016-10-04
Category : Mathematics
ISBN : 9781315355627

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Analytical Methods for Kolmogorov Equations by Luca Lorenzi Pdf

The second edition of this book has a new title that more accurately reflects the table of contents. Over the past few years, many new results have been proven in the field of partial differential equations. This edition takes those new results into account, in particular the study of nonautonomous operators with unbounded coefficients, which has received great attention. Additionally, this edition is the first to use a unified approach to contain the new results in a singular place.

Finite Element Methods for Eigenvalue Problems

Jiguang Sun,Aihui Zhou
Finite Element Methods for Eigenvalue Problems Book PDF

Author : Jiguang Sun,Aihui Zhou
Publisher : CRC Press
Page : 368 pages
File Size : 40,9 Mb
Release : 2016-08-19
Category : Mathematics
ISBN : 9781482254655

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Finite Element Methods for Eigenvalue Problems by Jiguang Sun,Aihui Zhou Pdf

This book covers finite element methods for several typical eigenvalues that arise from science and engineering. Both theory and implementation are covered in depth at the graduate level. The background for typical eigenvalue problems is included along with functional analysis tools, finite element discretization methods, convergence analysis, techniques for matrix evaluation problems, and computer implementation. The book also presents new methods, such as the discontinuous Galerkin method, and new problems, such as the transmission eigenvalue problem.

Bounds for Determinants of Linear Operators and their Applications

Michael Gil'
Bounds for Determinants of Linear Operators and their Applications Book PDF

Author : Michael Gil'
Publisher : CRC Press
Page : 153 pages
File Size : 44,8 Mb
Release : 2017-03-03
Category : Mathematics
ISBN : 9781351652315

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Bounds for Determinants of Linear Operators and their Applications by Michael Gil' Pdf

This book deals with the determinants of linear operators in Euclidean, Hilbert and Banach spaces. Determinants of operators give us an important tool for solving linear equations and invertibility conditions for linear operators, enable us to describe the spectra, to evaluate the multiplicities of eigenvalues, etc. We derive upper and lower bounds, and perturbation results for determinants, and discuss applications of our theoretical results to spectrum perturbations, matrix equations, two parameter eigenvalue problems, as well as to differential, difference and functional-differential equations.

Integration and Cubature Methods

Willi Freeden,Martin Gutting
Integration and Cubature Methods Book PDF

Author : Willi Freeden,Martin Gutting
Publisher : CRC Press
Page : 501 pages
File Size : 47,9 Mb
Release : 2017-11-22
Category : Mathematics
ISBN : 9781351764766

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Integration and Cubature Methods by Willi Freeden,Martin Gutting Pdf

In industry and economics, the most common solutions of partial differential equations involving multivariate numerical integration over cuboids include techniques of iterated one-dimensional approximate integration. In geosciences, however, the integrals are extended over potato-like volumes (such as the ball, ellipsoid, geoid, or the Earth) and their boundary surfaces which require specific multi-variate approximate integration methods. Integration and Cubature Methods: A Geomathematically Oriented Course provides a basic foundation for students, researchers, and practitioners interested in precisely these areas, as well as breaking new ground in integration and cubature in geomathematics.

Elements of Quasigroup Theory and Applications

Victor Shcherbacov
Elements of Quasigroup Theory and Applications Book PDF

Author : Victor Shcherbacov
Publisher : CRC Press
Page : 423 pages
File Size : 51,8 Mb
Release : 2017-05-12
Category : Computers
ISBN : 9781351646369

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Elements of Quasigroup Theory and Applications by Victor Shcherbacov Pdf

This book provides an introduction to quasigroup theory along with new structural results on some of the quasigroup classes. Many results are presented with some of them from mathematicians of the former USSR. These included results have not been published before in the western mathematical literature. In addition, many of the achievements obtained with regard to applications of quasigroups in coding theory and cryptology are described.

Delay Differential Evolutions Subjected to Nonlocal Initial Conditions

Monica-Dana Burlică,Mihai Necula,Daniela Roșu,Ioan I. Vrabie
Delay Differential Evolutions Subjected to Nonlocal Initial Conditions Book PDF

Author : Monica-Dana Burlică,Mihai Necula,Daniela Roșu,Ioan I. Vrabie
Publisher : CRC Press
Page : 362 pages
File Size : 46,6 Mb
Release : 2018-09-03
Category : Mathematics
ISBN : 9781498746465

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Delay Differential Evolutions Subjected to Nonlocal Initial Conditions by Monica-Dana Burlică,Mihai Necula,Daniela Roșu,Ioan I. Vrabie Pdf

Filling a gap in the literature, Delay Differential Evolutions Subjected to Nonlocal Initial Conditions reveals important results on ordinary differential equations (ODEs) and partial differential equations (PDEs). It presents very recent results relating to the existence, boundedness, regularity, and asymptotic behavior of global solutions for differential equations and inclusions, with or without delay, subjected to nonlocal implicit initial conditions. After preliminaries on nonlinear evolution equations governed by dissipative operators, the book gives a thorough study of the existence, uniqueness, and asymptotic behavior of global bounded solutions for differential equations with delay and local initial conditions. It then focuses on two important nonlocal cases: autonomous and quasi-autonomous. The authors next discuss sufficient conditions for the existence of almost periodic solutions, describe evolution systems with delay and nonlocal initial conditions, examine delay evolution inclusions, and extend some results to the multivalued case of reaction-diffusion systems. The book concludes with results on viability for nonlocal evolution inclusions.

Iterative Methods and Preconditioning for Large and Sparse Linear Systems with Applications

Daniele Bertaccini,Fabio Durastante
Iterative Methods and Preconditioning for Large and Sparse Linear Systems with Applications Book PDF

Author : Daniele Bertaccini,Fabio Durastante
Publisher : CRC Press
Page : 375 pages
File Size : 55,7 Mb
Release : 2018-02-19
Category : Mathematics
ISBN : 9781498764179

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Iterative Methods and Preconditioning for Large and Sparse Linear Systems with Applications by Daniele Bertaccini,Fabio Durastante Pdf

This book describes, in a basic way, the most useful and effective iterative solvers and appropriate preconditioning techniques for some of the most important classes of large and sparse linear systems. The solution of large and sparse linear systems is the most time-consuming part for most of the scientific computing simulations. Indeed, mathematical models become more and more accurate by including a greater volume of data, but this requires the solution of larger and harder algebraic systems. In recent years, research has focused on the efficient solution of large sparse and/or structured systems generated by the discretization of numerical models by using iterative solvers.