Geometric Numerical Integration

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Release : 2013-03-09
Genre : Mathematics
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Book Rating : 188/5 ( reviews)

Geometric Numerical Integration - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Geometric Numerical Integration write by Ernst Hairer. This book was released on 2013-03-09. Geometric Numerical Integration available 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.

A Concise Introduction to Geometric Numerical Integration

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Release : 2017-11-22
Genre : Mathematics
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Book Rating : 861/5 ( reviews)

A Concise Introduction to Geometric Numerical Integration - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook A Concise Introduction to Geometric Numerical Integration write by Sergio Blanes. This book was released on 2017-11-22. A Concise Introduction to Geometric Numerical Integration available in PDF, EPUB and Kindle. 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.

Simulating Hamiltonian Dynamics

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Release : 2004
Genre : Mathematics
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Book Rating : 907/5 ( reviews)

Simulating Hamiltonian Dynamics - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Simulating Hamiltonian Dynamics write by Benedict Leimkuhler. This book was released on 2004. Simulating Hamiltonian Dynamics available 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.

Symplectic Geometric Algorithms for Hamiltonian Systems

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Release : 2010-10-18
Genre : Mathematics
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Book Rating : 770/5 ( reviews)

Symplectic Geometric Algorithms for Hamiltonian Systems - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Symplectic Geometric Algorithms for Hamiltonian Systems write by Kang Feng. This book was released on 2010-10-18. Symplectic Geometric Algorithms for Hamiltonian Systems available in PDF, EPUB and Kindle. "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.

A First Course in the Numerical Analysis of Differential Equations

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Release : 2009
Genre : Mathematics
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Book Rating : 908/5 ( reviews)

A First Course in the Numerical Analysis of Differential Equations - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook A First Course in the Numerical Analysis of Differential Equations write by A. Iserles. This book was released on 2009. A First Course in the Numerical Analysis of Differential Equations available 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.