Structure-Preserving Algorithms for Oscillatory Differential Equations II

ISBN-13: 9783662481554 / Angielski / Twarda / 2016 / 298 str.

Xinyuan Wu; Kai Liu; Wei Shi

Opis

This book describes a variety of highly effective and efficient structure-preserving algorithms for second-order oscillatory differential equations. The book describes novel advances in RKN methods, ERKN methods, Filon-type asymptotic methods, AVF methods, and trigonometric Fourier collocation methods.

Recenzje

"This monograph is an excellent reference for practicing scientists and engineers who need in-depth information about structure-preserving integration of oscillatory ODEs. Senior undergraduate and graduate students of applied mathematics, engineering and physics will find the book to be an invaluable resource. It can also be used as a textbook in courses on structure preserving numerical algorithms for IVPs." (Martin Hermann, Mathematical Reviews, May, 2017)

"The textbook demonstrates extensions and specialisations of numerical methods for the class of oscillatory problems given by second-order ordinary differential equations. This collection of scientific work is suitable for researchers as well as postgraduate students (Ph.D. students) in the field of numerical analysis." (Roland Pulch, zbMATH 1352.65187, 2017)

Spis treści

Matrix-variation-of-constants formula.- Improved St ¨ormer-Verlet formulae with applications.- Improved Filon-type asymptotic methods for highly oscillatory differential equations.- Efficient energy-preserving integrators for multi-frequency oscillatory Hamiltonian systems.- An extended discrete gradient formula for multi-frequency oscillatory Hamiltonian systems.- Trigonometric Fourier collocation methods for multi-frequency oscillatory systems.- Error bounds for explicit ERKN integrators for multi-frequency oscillatory systems.- Error analysis of explicit TSERKN methods for highly oscillatory systems.- Highly accurate explicit symplectic ERKN methods for multi-frequency oscillatory Hamiltonian systems.- Multidimensional ARKN methods for general multi-frequency oscillatory second-order IVPs.- A simplified Nystr¨om-tree theory for ERKN integrators solving oscillatory systems.- An efficient high-order explicit scheme for solving Hamiltonian nonlinear wave equations.

O autorze

Xinyuan Wu, professor in mathematics, Nanjing University, received his PhD degree in mathematics form Nanjing university. His research areas include numerical differential equations, numerical algebra, etc. Jianlin Xia, professor in mathematics, Purdue University, received his PhD degree in mathematics form University of California, Berkeley. His research areas include large scale linear systems and least squares problems, large eigenvalue problems, fast and reliable condition estimation, numerical differential equations, nonlinear systems.

Blurb

This book describes a variety of highly effective and efficient structure-preserving algorithms for second-order oscillatory differential equations. Such systems arise in many branches of science and engineering, and the examples in the book include systems from quantum physics, celestial mechanics and electronics. To accurately simulate the true behavior of such systems, a numerical algorithm must preserve as much as possible their key structural properties: time-reversibility, oscillation, symplecticity, and energy and momentum conservation. The book describes novel advances in RKN methods, ERKN methods, Filon-type asymptotic methods, AVF methods, and trigonometric Fourier collocation methods.  The accuracy and efficiency of each of these algorithms are tested via careful numerical simulations, and their structure-preserving properties are rigorously established by theoretical analysis. The book also gives insights into the practical implementation of the methods.

This book is intended for engineers and scientists investigating oscillatory systems, as well as for teachers and students who are interested in structure-preserving algorithms for differential equations.