\summary{Several time discretized domain for the incompressible Navier-Stokes equations (iNSE) in moving domains have been proposed in literature. Here, we introduce a unified formulation that combines different approaches found in literature, allowing a common well posedness and time stability analysis. It can be therefore shown that only a particular choice of numerical schemes ensure such properties under some restrictions. The analysis will be shown for Chorin-Temam schemes using the insight found in the monolithic case. Results are supported from numerical simulations and its usage in fluid-solid interaction problems in cardiac geometries will be presented.}
Several works have been reported dealing with numerical solutions of the iNSE in moving domains within the Arbitrary Lagrangian Eulerian formulation (ALE), primarily in the context of fluid-solid coupling, e.g. \cite{astorino-chouly-fernandez-09, bertoglio2013sisc}. Different choices of time discretization schemes have been reported e.g. \cite{Basting2017, Hessenthaler2017}, nevertheless to the best of the authors knowledge, only a few monolithic schemes have been throughly analyzed, e.g. \cite{Lozovskiy2018, smaldone2014, le-tallec-mouro-01, Burtschell2017} while no analysis has been reported for Chorin-Temam (CT) schemes, being an alternative option when requirements such a low time computations are needed, specially for future industrial applications.
The goal of this talk is to present the finding of well-posedness and unconditional energy balance of the iNSE-ALE for several reported CT discretization schemes within a single formulation, published in \cite{arostica2021monolithic}. The main result to show will be that under appropiate conditions on the rate of domain deformation, a first order time discretization scheme for the CT scheme is unconditionally stable.
Our finding will be supplemented with an application case of fluid-solid interactions problem in an idealize geometry, exploiting the splitting nature of the CT scheme with a well-known coupling approach \cite{bertoglio2013sisc}.