Titlebook: Geometric Numerical Integration; Structure-Preserving Ernst Hairer,Gerhard Wanner,Christian Lubich Book 2006Latest edition Springer-Verlag

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Oscillatory Differential Equations with Varying High Frequencies,inate transforms that separate slow and fast motions and relate the fast oscillations to the skew-hermitian linear case. For the numerical treatment we consider suitably constructed long-time-step methods (“adiabatic integrators”) and multiple time-stepping methods.

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Examples and Numerical Experiments,ffects (on a different scale) occur with more sophisticated higher-order integration schemes. The experiments presented here should serve as a motivation for the theoretical and practical investigations of later chapters. The reader is encouraged to repeat the experiments or to invent similar ones.

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Numerical Integrators,lly all high-order implicit Runge–Kutta methods of interest. We then treat partitioned Runge–Kutta methods and Nystr?m methods, which can be applied to partitioned problems such as Hamiltonian systems. Finally we present composition and splitting methods.

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Order Conditions, Trees and B-Series,recently found interesting applications in quantum field theory. The chapter terminates with the Baker- Campbell-Hausdorff formula, which allows another access to the order properties of composition and splitting methods.

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Oscillatory Differential Equations with Constant High Frequencies,been proposed in the literature – some of them decades ago, some very recently, motivated by problems from molecular dynamics, astrophysics and nonlinear wave equations. For these methods it is not obvious what implications geometric properties like symplecticity or reversibility have on the long-time behaviour, e.g., on energy conservation.

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Arbeitsbereich und Datenausgabe,hem. In particular, we study projection methods and methods based on local coordinates of the manifold defined by the invariants. We discuss in some detail the case where the manifold is a Lie group. Finally, we consider differential equations on manifolds with orthogonality constraints, which often arise in numerical linear algebra.