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 andcombinatorial 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.

Contains articles from highly distinguished experts in numerical analysis

Preface.- A. Iserles and G.R.W. Quispel, Why geometric numerical integration?.- B. Owren, Lie group integrators.- H. Z. Munthe-Kaas and K. K. Føllesdal, Lie-Butcher series, Geometry, Algebra and Computation.- A. Murua and J. M. Sanz-Serna, Averaging and computing normal forms with word series algorithms.- L. A. Duffaut Espinosa, K. Ebrahimi-Fard, and W. Steven Gray, Combinatorial Hopf algebras for interconnected nonlinear input-output systems with a view towards discretization.- F. Casas, Computational aspects of some exponential identities.- K. Ebrahimi-Fard and I. Mencattini, Post-Lie Algebras, Factorization Theorems and Isospectral Flows.- G. Bogfjellmo, R. Dahmen, and A.Schmeding, Overview of (pro-)Lie group structures on Hopf algebra character groups,.- M. Barbero Liñán and D. Martín de Diego, Bäcklund transformations in discrete variational principles for Lie-Poisson equations.- M. Vermeeren, Numerical precession in variational discretizations of the Kepler problem.- O. Verdier, Full affine equivariance and weak natural transformations in numerical analysis - the case of B-Series.- References.