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MaGIC 2000, Ustaoset: List of abstracts

February 21-25

Kenth Engø

Time symmetry of Crouch-Grossmann methods

(Approx 30 min)

We discuss time symmetry of CG-methods by introducing and motivating the adjoint method of a CG-method.

Stig Faltinsen

Geometric integration of Hamiltonian problems on Lie groups

(Approx 45-50 min)

In this talk we present a unified approach to approximating different variants of Hamiltonian problems on Lie groups using Lie-group methods. We extend the joint work with Engø and show how discrete gradients combined with different actions gives Casimir and energy preserving methods. Furthermore we address the question of whether Lie-group methods can be symplectic (or Lie--Poisson). First of all we show that the criteria for symplecticity for the Lie--Poisson and right-trivialised equations are equivalent for Lie-group methods. The symplectic method of Lewis and Simo for the rigid body is in this formulism a variant of the Lie trapezoidal rule. We show in this paper how one must alter the Lie midpoint method in order to make it symplectic for some low-dimensional Lie algebras. Numerical experiments are shown to illustrate our claims.

Hans Munthe-Kaas

On some application of geometric spaces in G.I.

(Approx 45 min)

Symmetric spaces are fundamental in differential geometry. They are defined as Riemannian manifolds where all isometries preserve the Riemann tensor, hence these spaces have constant curvature. It turns out that symmetric spaces have many interesting applications in Geometric Integration, such as the classification of sandwich products (a la McLachlan/Quispel), generalizing polar decompositions, the study of time reversal symmetry and providing a framework for studying numerical integrators on symmetric spaces (e.g. the space of all symmetric positive definite matrices). We will give an introductions to these topics, and hopefully present some algorithms.

This talk provides background theory for that of Antonella Zanna

Roman Kozlov

Conservation laws of semidiscrete Schrödinger equation

(Approx. 45 min)

Click here for the abstract (.ps file)

Antonella Zanna

Generalization of the polar decomposition to symmetric spaces

(Approx. 45 min)

It is well known in linear algebra that an arbitrary real matrix A can be written as A=HU, where H is a symmetric positive (semi-)definite matrix and U is orthogonal. Furthermore U is the best orthogonal approximant to A in any orthogonal invariant norm. In this talk we shall generalize the above concept to elements in a Lie group, provided they are sufficiently close to the identity. Thus, given an involutive automorphism \sigma, any Lie-group element z can be written as z=xy, where x is in a symmetric space and y in the subgroup of fixed points of \sigma. The best approximant properties of y are also retained in any right-invariant norm.

This talk requires backgrount theory from Hans Munthe-Kaas

Krister Åhlander

Explicit Runge--Kutta methods as state machines

(Approx. 30 min)

We discuss the implementation of explicit Runge--Kutta methods as state machines and why this approach is useful in the context of PDEs.