1st Edition
A Concise Introduction to Geometric Numerical Integration
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
Introduction
Composition and splitting
Order conditions of splitting and composition methods
Splitting methods for special systems
Processing
Splitting methods for non-autonomous systems
A collection of low order splitting and composition methods
Illustrations
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
Introduction
Splitting methods for the time-dependent Schrödinger equation
Splitting methods for parabolic evolution equations
Appendix: Some additional mathematical results
Bibliography
Index
Exercises appear at the end of each chapter.
Biography
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
