Waveform methods for partitioned dynamic coupling (Birken Philippe, University of Lund)

18.02.2025 14:00

We consider coupled time dependent partial differential equations on separate domains, This can be used to model conjugate heat transfer, but also the transfer of wind stress between ocean and atmosphere, or flutter of airplanes. Our interest is in so called waveform relaxation. There, the subproblems are solved iteratively on a time window, given an approximation of the solution to the other problem. These methods open up variable time steps and time adaptivity while using separate codes for the subproblem. They can be combined with a relaxation step or Quasi-Newton acceleration.

We first present a novel time adaptive Quasi-Newton waveform method. We then discuss convergence theory of this method. This is quite limited for the general case. For the simpler case of linear coupled heat equation with discontinuous coefficients, much more can be said in dependence on the material parameters, the mesh width and the time steps. Over the years, this has been approached on different levels, meaning fully discrete, semidiscrete, or fully continuous. Thereby, one has obtained both superlinear convergence results, and norm estimates. Currently, these results give an incomplete and seemingly conflicting picture. We review these results, and then fill some of the gaps.

This is joint work with Niklas Kotarsky (Lund) and Martin Gander (Geneva).

Lieu

Conseil Général 7-9, Room 1-05, Séminaire d'analyse numérique

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