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Stability and error control for higher order in time ale formulations

  • Date May 7, 2015
  • Hour 3 pm
  • Room Room D
  • Speaker Irene Kyza (University of Dundee)

Arbitrary Lagrangian Eulerian (ALE) formulations are useful when dealing with problems dened on deformable domains, such as uid{structure interactions. Having at hand higher order (at least second order) in time ALE methods is important, as they lead to realistic simulations involving uids in 3d. However such methods are very limited in the literature.

In this talk we present some recent results on the design, the stability and the error control of discontinuous Galerkin (dG) ALE methods of any order, for an evolution convection-diusion model problem on time-dependent domains.

Exploiting the variational structure of the dG method we prove that our dG schemes enjoy the same stability properties as the continuous problem. The same is true for the practical Reynolds' methods which result from the dG methods after numerical integration by quadratures which inherit a discrete Reynolds' identity. This is a generalization of the so-called Geometric Conservation Law (GCL) to higher order methods. We also study the stability properties of Runge-Kutta-Radau methods; these methods are proven to be stable under a mild constraint on the time-step related to the motion of the domain.

Finally, using the stability properties of the methods, we provide a priori and a posteriori error analyses. The a priori error analysis also uses a novel time{space projection, the so-called \ALE projection", while the a posteriori error analysis is based on the denition of a novel time{space reconstruction.