Section: New Results
High order embedded and immersed boundary methods
Immersed and embedded methods are a flexible and efficient approach to handle complex, and moving geometries. In these techniques, domain boundaries are not meshed exactly, but are embedded/immersed in the grid. Ad-hoc techniques are then required to account for the effect of the coupling/boundary conditions defined on the domain boundaries on the nodes in the computational domain. This year we have made some progress on these methods on two aspects.
Preliminary results on the use of higher order schemes with penalization have been obtained. As in , the main idea is to combine a simple penalization technique to impose the no-slip condition with mesh adaptation to control the error. In the work done this year we have looked into the possibility of exploiting curved meshes and higher order schemes to further improve the accuracy in laminar flow computations , .
In parallel, we have worked on a different strategy to achieve higher order in the context of an embedded approach: the shifted boundary method, initially proposed in , . We have worked on an extension of this method to hyperbolic problems, which are difficult to treat with penalization. The extension proposed for linear waves, as well as for the nonlinear shallow water equations show that second order results can be easily achieved in this context  (see also , ).