A Concise Introduction to Geometric Numerical Integration: 1st Edition (Hardback) book cover

A Concise Introduction to Geometric Numerical Integration

1st Edition

By Sergio Blanes, Fernando Casas

Chapman and Hall/CRC

218 pages | 23 B/W Illus.

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pub: 2016-05-23
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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] is highly recommended for graduate students, postgraduate researchers, and researchers interested in beginning study in the field of geometric numerical integration."

—David Cohen, Mathematical Reviews, November 2017

Table of Contents

What is geometric numerical integration?

First elementary examples and numerical methods

Classical paradigm of numerical integration

Towards a new paradigm: geometric numerical integration

Symplectic integration

Illustration: the Kepler problem

What is to be treated in this book (and what is not)

Classical integrators and preservation of properties

Taylor series methods

Runge–Kutta methods

Multistep methods

Numerical examples

Splitting and composition methods


Composition and splitting

Order conditions of splitting and composition methods

Splitting methods for special systems


Splitting methods for non-autonomous systems

A collection of low order splitting and composition methods


Other types of geometric numerical integrators

Symplectic methods based on generating functions

Variational integrators

Volume-preserving methods

Lie group methods

Long-time behavior of geometric integrators

Introduction. Examples

Modified equations

Modified equations of splitting and composition methods

Estimates over long-time intervals

Application: extrapolation methods

Time-splitting methods for PDEs of evolution


Splitting methods for the time-dependent Schrödinger equation

Splitting methods for parabolic evolution equations

Appendix: Some additional mathematical results



Exercises appear at the end of each chapter.

About the Authors

Sergio Blanes is an associate professor of applied mathematics at the Universitat Politècnica de València. He is also editor of The Journal of Geometric Mechanics. He was a postdoc researcher at the University of Cambridge, University of Bath, and University of California, San Diego. His research interests include geometric numerical integration and computational mathematics and physics.

Fernando Casas is a professor of applied mathematics at the Universitat Jaume I. His research focuses on geometric numerical integration, including the design and analysis of splitting and composition methods for differential equations and their applications, Lie group methods, perturbation techniques, and the algebraic issues involved.

About the Series

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Arithmetic
MATHEMATICS / Differential Equations
MATHEMATICS / Number Systems