The LS-STAG method: A new immersed boundary/level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties

The fixed-mesh ALE approach for the numerical approximation of flows in moving domains

“…Note that the order of accuracy of this approximation is consistent with the order of accuracy of (4)- (5), that is to say, it is 2 for θ = 1/2 and 1 otherwise. We are interested only in the cases θ = 1/2 and θ = 1 (implicit schemes are required).…”

“…As an example, in [35] the authors extrapolate the velocity and pressure from the nearest fluid nodes at the previous time step. In [5], the Navier-Stokes equations are correctly expressed in an ALE framework, but the velocity is taken as the solid velocity. It is worth to note that if the solid is deformable and has been solved together with the fluid in a coupled way (as in the original immersed boundary method [40] or in the fluid-solid approach in [46]), this velocity is physically meaningful.…”

“…The next step is to consider the spatial discretization of problem (4)- (5). As for the time discretization, different options are possible.…”

“…It is based on the subgrid scale concept introduced in [28], although when linear elements are used it reduces to the Galerkin/leastsquares method described for example in [18]. We apply this stabilized formulation together with the finite difference approximation in time (4)- (5).…”

“…In our case, this operator corresponds to the linearized form of the time discrete Navier-Stokes equations (4)- (5). In this case, the method consists of finding u …”