MaGIC 2000, Ustaoset: List of abstracts
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.
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.
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.
Conservation laws of semidiscrete Schrödinger equation(Approx. 45 min)
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.
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.