1st Edition

A Concise Introduction to Geometric Numerical Integration

By Sergio Blanes, Fernando Casas Copyright 2016
    232 Pages 23 B/W Illustrations
    by Chapman & Hall

    230 Pages 23 B/W Illustrations
    by Chapman & Hall

    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.

    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.


    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.

    "[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