Numerical Integration with Splitting Methods

Alexander Ostermann

Institut für Mathematik, Universität Innsbruck

A mini course at Helsinki University of Technology,
27.-30.04.2009

A part of the Special Year in Numerics 2008-2009
of the Finnish Mathematical Society



Splitting methods constitute a class of (numerical) schemes for solving initial value problems. Roughly speaking, they decompose the vector field into several parts and integrate these parts separately. Splitting methods are frequently used for the time discretisation of partial differential equations. The reason for this is that the splitting procedure yields highly competitive time stepping schemes which can dramatically reduce the required computational effort, compared to schemes based on the full vector field.

The first part of the mini-course will be devoted to splitting methods for ODEs. We shall derive the (non-stiff) order conditions via the well-known Baker-Campbell-Hausdorff formula.

In the second part, emphasis will be laid on a sound convergence theory for splitting methods applied to time dependent partial differential equations. In particular, we will cover the following topics:
  • evolution equations with unbounded operators;
  • splitting methods for Schrödinger equations;
  • dimension splitting for parabolic problems;
  • boundary conditions;
  • high-order splitting methods for analytic semigroups;
  • nonlinear problems.
References can be found here.

Lectures:

  • Monday 27.4.2009 at 10:15-12:00 in room U356,
  • Tuesday 28.4.2009 at 10:15-12:00 in room U358,
  • Wednesday 29.4.2009 at 10:15-12:00 in room U322,
  • Thursday 30.4.2009 at 10:15-12:00 in room U322.

Computer practice sessions:

  • Monday 27.4.2009 at 14:15-16:00 in room Y339a,
  • Tuesday 28.4.2009 at 14:15-16:00 in room Y339b,
  • Wednesday 29.4.2009 at 14:15-16:00 in room Y338c.
Contact information:
Professor Timo Eirola (Timo.Eirola@tkk.fi)
Assistant Kurt Baarman (Kurt.Baarman@tkk.fi)