MaGIC 2002, Ustaoset: Talks and abstracts

February 4-8

• Borko Minchev:
  "The order of exponential polar-type splitting for some stiff problems"
  (45 min)

  Abstract: Splitting method for the approximation of the exponential is used in many applications. It is known that for symmetric splitting schemes in case of stiff and nonstiff parts of ODE one may observe an order reduction of the splitting method for sufficiently large time steps. Similar result can be observed if we use exponential splitting based on the generalized polar decomposition (polar-type splitting).

• Elena Celledoni:
  "The double-pendulum equation: modelling and numerical methods"
  (45 min)
• Bjarte Hagland:
  TBA
  (30-45 min)
• Stein Krogstad:
  "On the computation of Lyapunov exponents for Hamiltonian and reversible systems"
  (45-60 min)
• Hans Munthe-Kaas:
  "On equivariance and symmetry of numerical integrators"
  (60 min)

  (Report on 'work that should have been in progress')
  Abstract: In our papers we often bring forward the (obvious) point that Lie group integrators 'stay on the correct manifold'. Other authors, e.g. Hairer, make a point of explaining that staying on a manifold is not very interesting unless one also have additional properties of the integrators. He is pointing out things like time symmetry.
  An other property that was implicit in early work (but not explicitly stated!) was the property of equivariance, the numerical integrator should respect the motions of the underlying group, e.g. Rotate * numerical integration = numerical integration * rotation. This is a property that is shared by all our early integrators, such as e.g. the CG methods, and exponential methods. For the "Kenth class of algorithms", discussed by Kenth in his BiT article (general mappings liealg->liegroup + Lie group action), equivariance is achieved if we put some conditions on allowed maps from algebra to group. In general methods based on local coordinates or retractions, the equivariance is not necessarily preserved. We will in this talk discuss equivariance and its implications for quality of numerics. Furthermore, we will arrive at some resuts that are surprising, at least to me. Here is one: *** Retraction method + equivariance => method is an equivariant Kenth class algorithm **** The opposite implication does, however, not hold. So if we require equivariance, we have retraction methods < Kenth class. Still it is useful to consider retraction methods as a class of methods in their own right, since the inclusion above is not practical from a computational point of view.

• Jitse Niesen:
  "On the global error committed by numerical integrators on nonautonomous oscillator"
  (60 min)

  Abstract: In this talk, we show how to obtain a priori estimates for the global error of a numerical integrator. Our approach is based on perturbation theory and modified equations. The estimates are correct up to a term of order h^{2p}, where h denotes the (constant) step size and p is the order of the method.
  We apply the resulting estimates to two classes of nonautonomous oscillators. The first class consists of linear oscillators which are amenable to the Liouville-Green approximation (this work is mainly due to Arieh Iserles). The second class is formed by nonlinear oscillators satisfying the Emden-Fowler equation y'' + t^v y^n = 0. We note some substantial differences between both classes during the computation. Numerical results confirm the correctness of our estimates.

• Brynjulf Owren:
  "Expansions of Lie-group methods and their algebraic structure"
  (60 min)
• Bård Skaflestad:
  TBA
  (30-45 min)
• Andrzej Suslovic:
  TBA
  (60 min)
• Antonella Zanna:
  "On global bounds for matrix exponential splittings based on Generalized Polar Decompositions"
  (45 min)
• Krister Ålander:
  "An overview of EinSum symmetries and band tensor notation"
  (60 min)

  Abstract: EinSum is a C++ package which understands index notation including the Einstein summation convention. In this talk we will focus on its support for tensor symmetries, and we will also discuss generalizations to "band tensors".