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This volume resulted from presentations given at the international “Brainstorming Workshop on New Developments in Discrete Mechanics, Geometric Integration and Lie–Butcher Series”, that took place at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid, Spain. It combines overview and research articles on recent and ongoing developments, as well as new research directions.
Why geometric numerical integration? In their article of the same title Arieh Iserles and Reinout Quispel, two renowned experts in numerical analysis of differential equations, provide a compelling answer to this question. After this introductory chapter a collection of high-quality research articles aim at exploring recent and ongoing developments, as well as new research directions in the areas of geometric integration methods for differential equations, nonlinear systems interconnections, and discrete mechanics. One of the highlights is the unfolding of modern algebraic and combinatorial structures common to those topics, which give rise to fruitful interactions between theoretical as well as applied and computational perspectives.
The volume is aimed at researchers and graduate students interested in theoretical and computational problems in geometric integration theory, nonlinear control theory, and discrete mechanics.
Why Geometric Numerical Integration?
Arieh Iserles and G.R.W. Quispel
Lie Group Integrators
Brynjulf Owren
Lie–Butcher Series, Geometry, Algebra and Computation
Hans Z. Munthe-Kaas and Kristoffer K. Føllesdal
Averaging and Computing Normal Forms with Word Series Algorithms
Ander Murua and Jesús M. Sanz-Serna
Combinatorial Hopf Algebras for Interconnected Nonlinear Input-Output Systems with a View Towards Discretization
Luis A. Duffaut Espinosa, Kurusch Ebrahimi-Fard and W. Steven Gray
Computational Aspects of Some Exponential Identities
Fernando Casas
Post-Lie Algebras, Factorization Theorems and Isospectral Flows
Kurusch Ebrahimi-Fard and Igor Mencattini
Overview of (pro-)Lie Group Structures on Hopf Algebra Character Groups
Geir Bogfjellmo, Rafael Dahmen and Alexander Schmeding Bäcklund
Transformations in Discrete Variational Principles for Lie-Poisson Equations
María Barbero Liñán and David Martín de Diego
Numerical Precession in Variational Discretizations of the Kepler Problem
Mats Vermeeren
Full Affine Equivariance and Weak Natural Transformations in Numerical Analysis — The Case of B-Series
Olivier Verdier