Line Integral Methods for Conservative Problems

Line Integral Methods for Conservative Problems explains the numerical solution of differential equations within the framework of geometric integration, a branch of numerical analysis that devises numerical methods able to reproduce (in the discrete solution) relevant geometric properties of the continuous vector field. The book focuses on a large set of differential systems named conservative problems, particularly Hamiltonian systems. Assuming only basic knowledge of numerical quadrature and Runge–Kutta methods, this self-contained book begins with an introduction to the line integral methods. It describes numerous Hamiltonian problems encountered in a variety of applications and presents theoretical results concerning the main instance of line integral methods: the energy-conserving Runge–Kutta methods, also known as Hamiltonian boundary value methods (HBVMs). The authors go on to address the implementation of HBVMs in order to recover in the numerical solution what was expected from the theory. The book also covers the application of HBVMs to handle the numerical solution of Hamiltonian partial differential equations (PDEs) and explores extensions of the energy-conserving methods. With many examples of applications, this book provides an accessible guide to the subject yet gives you enough details to allow concrete use of the methods. MATLAB codes for implementing the methods are available online.

Table of Contents:
A Primer on Line Integral Methods. Examples of Hamiltonian Problems. Analysis of Hamiltonian Boundary Value Methods (HBVMs). Implementing the Methods and Numerical Illustrations. Hamiltonian Partial Differential Equations. Extensions. Appendix. Bibliography. Index.

About the Author :
Luigi Brugnano is a full professor of numerical analysis and chairman of the mathematics courses in the Department of Mathematics and Informatics at the University of Firenze. He is a member of several journal editorial boards. His research interests include matrix conditioning/preconditioning, parallel computing, computational fluid dynamics, numerical methods, iterative methods, geometric integration, and mathematical modeling and software. Felice Iavernaro is an associate professor of numerical analysis in the Department of Mathematics at the University of Bari. His primary interests include the design and implementation of efficient methods for the numerical solution of differential equations, particularly for the simulation of dynamical systems with geometric properties.

Review :
"Line Integral Methods for Conservative Problems presents an overview of numerical techniques based around the discrete line integral approach for solving conservative differential equation problems, with a particular emphasis on Hamiltonian problems. It showcases a large number of illustrative Hamiltonian examples that would be especially useful to advanced undergraduate students. Its main focus is the introduction, construction, and implementation of energy-conserving Runge-Kutta methods, also known as Hamiltonian boundary value methods (HBVMs), pioneered by the authors. The monograph concludes with extensions to Hamiltonian partial differential equations and other problems, such as multi-invariant and general conservative problems. The exposition is concise and lucid. The many examples and clear proofs and discussion make this monograph very readable. It is a welcome and important addition to the growing class of research in the field of geometric integration." -Kevin Burrage, Professor of Computational Systems Biology, University of Oxford, UK, and Professor of Computational Mathematics, Queensland University of Technology, Australia "This clearly written book is a valuable contribution to the literature on geometric numerical methods. The accompanying MATLAB software and the detailed treatment of Hamiltonian partial differential equations are strong features that will increase the usefulness of the monograph." -Professor Jesus Maria Sanz-Serna, Universidad Carlos III de Madrid and Real Academia de Ciencias, Spain "The recent emphasis on geometric Integration, and in particular on energy-preserving line integral methods, has made the subject of numerical methods for differential equations very wide ranging and there are relatively few people who have made contributions to more than a small part of this extensive subject. Each of the two authors of this new book is an expert in both practical and theoretical aspects of the subject and each is an excellent expositor. The final result is a comprehensive work that will be accessible to a large range of computational mathematicians. It presents the mathematics behind Hamiltonian mechanics and the line integral method both for experts as well as for practical users and, at the same time, shows how good algorithms are constructed with the utmost respect for the underlying theory." -John Butcher, Emeritus Professor, University of Auckland