On monolithic and Chorin-Temam schemes for incompressible flows in moving domains

Several time discretization schemes for the incompressible Navier-Stokes equations (iNSE) in moving domains have been proposed. Here we introduce them in a uniﬁed fashion, allow-ing a common well possedness and time stability analysis. It can be therefore shown that only a particular choice of the numerical scheme ensures such properties. The analysis is performed for monolithic and Chorin-Temam schemes. Results are supported by numerical experiments.


Introduction
Several works have been reported dealing with the numerical solution of the iNSE in moving domains within an Arbitrary Lagrangian Eulerian formulation (ALE), primarily in the context of fluid-solid coupling. In particular different choices of time discretization have been reported on [2,6,[9][10][11][13][14][15]17,19]. To the best of the authors knowledge, only a few monolithic schemes have been thoroughly analyzed, e.g. in [4,14,17,19], while no analysis has been reported for Chorin-Temam (CT) methods. The goal of this work is therefore to assess well-posedness and unconditional energy balance of the iNSE-ALE for all reported monolithic and CT discretization schemes within a single formulation.
The reminder of this paper is structured as follows: Section 2 provides the continuous problem that will be studied. Section 3 introduces a general monolithic scheme that characterizes several approaches used in literature, well-posedness and energy stability are studied and discussed. Section 4 introduces the Chorin-Temam schemes where time stability is analyzed. Finally, Section 5 provides numerical examples testing our results.

The continuous problem
In the following, let us consider a domain Ω 0 ⊂ R d with d = 2, 3 and a deformation mapping X : R d ×R + → R d that defines the time evolving domain Ω t := X (Ω 0 , t). We assume X a C 1 mapping in both coordinates, 1-to-1 with C 1 inverse. We denote X ∈ R d the cartesian coordinate system in Ω 0 and x t := X (X, t) the one in Ω t , by F t := ∂x t ∂X the deformation gradient, H t := (F t ) −1 its inverse and J t := det(F t ) its Jacobian. Similarly, Grad(f ) := ∂f ∂X , Div(f ) := ∂ ∂X · f denote the gradient and divergence operators respectively and t (f ) := 1 2 (Grad(f )H t +(H t ) T Grad(f ) T ) the symmetric gradient, for f a well-defined vector function.
By H 1 0 (Ω 0 ) we denote the standard Sobolev space of vector fields u defined in Ω 0 with values in R d such that u = 0 on ∂Ω 0 , by L 2 0 (Ω 0 ) the standard square integrable space of functions r defined in Ω 0 with values in R s.t. Ω 0 r dX = 0 and T > 0 a final time. We consider the weak form of the iNSE in ALE form [16,Ch. 5] given initial and w := ∂X ∂t time-varying domain velocities. For the sake of simplicity, we omit the time-dependency on the fields u, p. Notice that the velocity flow at time t is given by u • X −1 (·, t).
(Ω 0 ) a solution of Problem (1), the following energy balance holds: In the general case with non-homogeneous Dirichlet boundary conditions, the energy balance also includes flow intensification due to the moving boundary. In such case, the intensification term appearing on the energy balance (2) in given by: where N ∈ R d denotes the outward normal.
Remark 3. Although Dirichlet boundary conditions are used throughout this work, it can be extended straightforwardly to the Neumann case by including the so called backflow stabilizations, see e.g. [3].
Remark 5. The extension of Proposition 2 to the case with non-homogeneous Dirichlet boundary conditions follows from the trace theorem by assuming Ω 0 a Lipschitz bounded open set [5].
• J n+1 + J n > 0 if = n, i.e. no restriction on the time step size, since we assume orientation preserving deformation mappings.

Proposition 4.
Under assumptions of Proposition 2 and α = β = 1, = n, the scheme (4) is unconditionally energy stable with energy estimate: Proof. By setting v = u n+1 in the bi-linear form (5), q = p n+1 in forms (6) and manipulating terms as standard in literature, the energy equality follows: Thus, for α = β = 1 and = n the result follows.
Remark 6. This works focuses on first-order schemes in time. The reason is that second order schemes, although stable in fixed domain, has been shown to be only conditionally stable in ALE form, as it was shown in [7] for the advection-diffusion problem for Crank-Nicolson (CN) and BDF (2). Therefore, we do not analyze here the schemes used in [6,9,18] -based on CN and used in the context of fluid-solid interaction -since their analysis repeats from [7]. Also in the same context, some authors have used the generalized α-methods since it is a popular scheme for elastodynamics [13]. However, there is no reported stability analysis even for the the fixed domain setting, and its stability properties are usually assumed to be transferred from the linear setting.
Proof. As standard in literature, let us take v =ũ n+1 in (FVS) n+1 , and q = p n in (PPS) n .
The results are assessed using time-dependent normalized parametersδ M := δ M /E st ,δ CT := δ CT /E st defined as: Figure 1 showsδ M ,δ CT values for each tested scheme. Propositions 4 and 5 are confirmed sinceδ M = 0 andδ CT ≤ 0 if = n. For = n + 1, peaks appearing throughout the simulation are defined by the sign change of domain velocity, i.e. in the change from expansion to contraction. Importantly, the spurious numerical energy rate related to discretization of the GCL condition appear to be positive in expansion, therefore being a potential source of numerical instabilities.

Conclusion
Several reported time discretization schemes for the iNSE-ALE have been reviewed and analyzed in terms of their well posedness at each time step and time stability. The stability analysis is confirmed by numerical experiments. For the monolithic case, two schemes lead to well-posed energy-stable problems whenever α = β = 1 with = n as studied in [14,17,19,20]. To the best of the authors knowledge, the unconditionally stable Chorin-Temam scheme derived in this work has not been reported yet.