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\title{On monolithic and Chorin-Temam schemes for incompressible flows in moving domains}

\author{
        Reidmen Ar\'{o}stica, Crist\'{o}bal Bertoglio \\
        Bernoulli Institute,   University of Groningen, 
        9747AG Groningen, The Netherlands
}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
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, allowing 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.
\end{abstract}

\section{Introduction}
%When working with flows from a simulation point of view, several schemes are proposed depending on its applications, suitable for specific requirement e.g. high spatio-temporal resolution, fidelity, stability or fast implementation/simulation times. 
%The literature is vast, but to our knowledge there is not a overview trying to summarize their approaches in a single scheme.

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 \cite{Basting2017, Murea2016, Landajuela2016,Lozovskiy2018,smaldone2014,langer2014numerical,LeTallec2001,Liu2018,failer2020impact,Hessenthaler2017}.
% In \cite{Basting2017, Murea2016, Landajuela2016} the FSI within Arbitrary Lagrangian-Eulerian (ALE) formalism in a linearized approach is studied, and \cite{Lozovskiy2018,smaldone2014} also include mass conservation terms. \cite{langer2014numerical,LeTallec2001,Liu2018} propose fully non-linear FSI schemes, and extensions using Crank-Nicholson are used by \cite{failer2020impact,Hessenthaler2017}. \CB{No entiendo la clasificacion de los papers, me parece que le falta un poco de coherencia. No deberian tambien haber mas papers, o estos son todos? }
%Thougthful 
To the best of the authors knowledge, only a few monolithic schemes have been thoroughly analyzed, e.g. in \cite{Lozovskiy2018, Burtschell2017, LeTallec2001, smaldone2014}, 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.
% Maybe we need to add the works of \cite{Boffi2004} for general surveys of ALE schemes.
%State-of-the art: monolithic (what is proven), CT (not proven, just "used") \\
%, which as it will be seen later on, holds under some restrictions with help of consistent stabilization terms. 
%The findings for the monolithic case are then used to introduce a Chorin-Temam scheme, for which unconditional energy stability holds. 

The reminder of this paper is structured as follows: 
Section \ref{sec:continuous_problem} provides the continuous problem that will be studied. Section \ref{sec:monolithic_schemes} introduces a general monolithic scheme that characterizes several approaches used in literature, well-posedness and energy stability are studied and discussed. Section \ref{sec:chorin_temam_schemes} introduces the Chorin-Temam schemes where time stability is analyzed. Finally, Section \ref{sec:numerical_examples} provides numerical examples testing our results.


\section{The continuous problem}
\label{sec:continuous_problem}

In the following, let us consider a domain $\Omega^0 \subset \mathbb{R}^d$ with $d = 2,3$ and a deformation mapping $\mathcal{X}: \mathbb{R}^d\times \mathbb{R}_{+}\mapsto\mathbb{R}^d$ that defines the time evolving domain $\Omega^t := \mathcal{X}(\Omega^0, t)$. 
We assume $\mathcal{X}$ a $\mathcal{C}^1$ mapping in both coordinates, 1-to-1 with $\mathcal{C}^1$ inverse. We denote $\mathbf{X} \in \mathbb{R}^d$ the cartesian coordinate system in $\Omega^0$ and $\mathbf{x}^t := \mathcal{X}(\mathbf{X}, t)$ the one in $\Omega^t$, by $F^t := \frac{\partial \mathbf{x}^t}{\partial \mathbf{X}}$ the deformation gradient, $H^t := (F^t)^{-1}$ its inverse and $J^t := det(F^t)$ its Jacobian. 
Similarly, $Grad(\mathbf{\mathbf{f}}) := \frac{\partial \mathbf{f}}{\partial \mathbf{X}}$, $Div(\mathbf{f}) := \frac{\partial}{\partial \mathbf{X}} \cdot \mathbf{f}$ denote the gradient and divergence operators respectively and $\epsilon^t(\mathbf{f}) := \frac{1}{2}( Grad(\mathbf{f}) H^t + (H^t)^{T} Grad(\mathbf{f})^{T})$ the symmetric gradient, for $\mathbf{f}$ a well-defined vector function. 

By $\mathbf{H}^1_0 (\Omega^0)$ we denote the standard Sobolev space of vector fields $\mathbf{u}$ defined in $\Omega^0$ with values in $\mathbb{R}^d$ such that $\mathbf{u} = \mathbf{0}$ on $\partial \Omega^0$, by $L^2_0(\Omega^0)$ the standard square integrable space of functions $r$ defined in $\Omega^0$ with values in $\mathbb{R}$ s.t. $\int_{\Omega^0} r \, \text{d}\mathbf{X} = 0$ and $T > 0$ a final time.
We consider the weak form of the iNSE in ALE form \cite[Ch. 5]{Richter2017}:  Find $(\mathbf{u}(t), p(t)) \in \mathbf{H}^{1}_{0}(\Omega^0) \times L^2_0(\Omega^0)$ for $t \in (0, T)$ with $\mathbf{u}(0) = \mathbf{u}_{init}$ s.t.
\begin{equation}
\label{eq:continuous_formulation}
\begin{aligned}
    \int_{\Omega^0} \rho J^t \frac{\partial \mathbf{u} }{\partial t} \cdot \mathbf{v} + \rho J^t Grad(\mathbf{u}) H^t (\mathbf{u} - \mathbf{w}) \cdot \mathbf{v} + J^t 2\mu \, \epsilon^t(\mathbf{u}):\epsilon^t(\mathbf{v})  \dif{X} & \\
      - \int_{\Omega^0} Div(J^t H^t \mathbf{v}) p \dif{X} + \int_{\Omega^0} Div(J^t H^t \mathbf{u})q \dif{X} &= 0%& \\
\end{aligned}
\end{equation}
for all $(\mathbf{v}, q) \in \mathbf{H}^1_0(\Omega^0) \times L^2_0(\Omega^0)$, $\mathbf{u}_{init} \in \mathbf{H}^1_0(\Omega^0)$ given initial and $\mathbf{w} := \frac{\partial \mathcal{X}}{\partial t}$ time-varying domain velocities. For the sake of simplicity, we omit the time-dependency on the fields $\mathbf{u}, p$.
Notice that the velocity flow at time $t$ is given by $\mathbf{u} \circ \mathcal{X}^{-1}(\cdot, t)$.


\begin{proposition}
\label{prop:energy_continuous} \cite[Chap. 9]{Quarteroni2009Cardiovascular}
Provided $(\mathbf{u}(t), p(t)) \in \mathbf{H}^1_0(\Omega^0) \times L^2_0(\Omega^0)$ a solution of Problem \eqref{eq:continuous_formulation}, % and $\mathbf{u}_{\mathbf{g}}(t) \in \mathbf{H}^1_{\mathbf{g}}(\Omega^0)$ the extension of $g \in L^2(\partial \Omega^0)$
the following energy balance holds:
\begin{equation}
\label{eq:continuous_energy_estimate}
\begin{aligned}
\frac{\partial}{\partial t}  \int_{\Omega^0} \frac{\rho}{2} J^t \vert \mathbf{u} \vert^2 \, \text{d}\mathbf{X}  =& - \int_{\Omega^0} J^t 2\mu \vert \epsilon^t (\mathbf{u}) \vert^2 \, \text{d}\mathbf{X}. %- \int_{\partial\Omega_0} \rho J_t F^{-1}_{t} (\mathbf{u} - \mathbf{w}) \cdot \mathbf{N} \, \text{d}\mathbf{S} + \mathcal{D}(\mathbf{u}; \mathbf{u}_{\mathbf{g}}) 
\end{aligned}
\end{equation}
\end{proposition}


\begin{remark}
Proposition \ref{prop:energy_continuous} makes use of the \textit{Geometric Conservation Law} (GCL) $\frac{\partial J^t}{\partial t} = Div\left( J^t F_{t}^{-1} \mathbf{w}\right)$. 
\end{remark}
\begin{remark}
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 \eqref{eq:continuous_energy_estimate} in given by:
\begin{equation}
\label{eq:extension_energy_continuous}
\int_{\partial\Omega^0} \rho \frac{\vert \mathbf{u} \vert^2}{2} J^t H^t (\mathbf{u} - \mathbf{w}) \cdot \mathbf{N} \, \text{d}\mathbf{S}
\end{equation}
where $\mathbf{N} \in \mathbb{R}^d$ denotes the outward normal.

\end{remark}
\begin{remark}
Although Dirichlet boundary conditions are used throughout this work, it can be extended straightforwardly to the Neumann case by including the so called \textit{backflow stabilizations}, see e.g. \cite{bertoglio2018benchmark}.
\end{remark}


\section{Monolithic schemes (first order in time)}
\label{sec:monolithic_schemes}

Most of the numerical schemes for Problem \eqref{eq:continuous_formulation} reported in the literature are first order and can be written as follows. 
Let $(t^n)_{n \in \mathbb{N}}$ be a uniform discretization of the time interval $(0, T)$ with step size $\tau > 0$
and let $H^n := H^{t^n}, J^n := J^{t^n}, w^n := w(t^n)$ be discrete sequences.
Given a conforming finite element space $\mathbf{V} \times Q$ of $\mathbf{H}^1_0(\Omega^0) \times L^2_0(\Omega^0)$ for velocity and pressure fields, the discrete problem of interest reads:  
Find $(\mathbf{u}^{n+1}, p^{n+1}) \in \mathbf{V} \times Q$ s.t.
\begin{equation}
\label{eq:discretized_monolithic_formulation}
\mathcal{A}(\mathbf{u}^{n+1},\mathbf{v}) - \mathcal{B}(\mathbf{v},p^{n+1}) + \mathcal{B}(\mathbf{u}^{n+1},q) = \mathcal{F}(\mathbf{v}) \quad \forall (\mathbf{v}, q) \in \mathbf{V} \times Q
\end{equation}
being
\begin{equation}
\label{eq:lhs_bilinear_form_A}
\begin{aligned}
\mathcal{A} (\mathbf{u}, \mathbf{v}) := & \int_{\Omega^0} \rho \frac{J^{\star\star}}{\tau} \mathbf{u} \cdot \mathbf{v} \, \text{d}\mathbf{X} + \int_{\Omega^0} \rho J^{\star} Grad(\mathbf{u}) H^{\star} (\mathbf{u}^{\ast} - \mathbf{w}^{\ast\ast}) \cdot \mathbf{v} \, \text{d}\mathbf{X} + \int_{\Omega^0} J^{\star} 2\mu \epsilon^{\star}(\mathbf{u}):\epsilon^{\star}(\mathbf{v}) \, \text{d}\mathbf{X} \\
& + \alpha \int_{\Omega^0} \frac{\rho}{2} \left( \frac{J^{n+1} - J^{n}}{\tau} - Div\left( J^{\star} H^{\star} \mathbf{w}^{\ast\ast} \right) \right) \mathbf{u} \cdot \mathbf{v} \, \text{d}\mathbf{X} + \beta \int_{\Omega^0} \frac{\rho}{2} Div\left( J^{\star} H^{\star} \mathbf{u}^{\ast} \right) \mathbf{u} \cdot \mathbf{v} \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
with $\alpha, \beta \in \{0, 1\}$ given parameters, and
%The matrix $B \in \mathbb{R}^{m^{Q}\times m^{\mathbf{V}}}$ and vector $F_n \in \mathbb{R}^n$ correspond also to the discretization of the bilinear and linear forms:
\begin{equation}
\label{eq:remaining_forms}
\begin{aligned}
\mathcal{B}(\mathbf{u}, q) & := \int_{\Omega^0} Div\left( J^{\star} H^{\star} \mathbf{u} \right) q \, \text{d}\mathbf{X} \quad \forall q \in Q, \quad \mathcal{F}(\mathbf{v}) := \int_{\Omega^0} \rho \frac{J^{\star\star}}{\tau} \mathbf{u}^n \cdot \mathbf{v} \, \text{d}\mathbf{X} \quad \forall \mathbf{v} \in \mathbf{V}
\end{aligned}
\end{equation}
\begin{remark}
The term multiplying  $\alpha$ is the discrete residual of GCL, while the one multiplying $\beta$ is a strongly consistent term vanishing for incompressible velocity fields.
\end{remark}


Formulation \eqref{eq:discretized_monolithic_formulation} contains a wide family of reported methods:
\begin{itemize}
\item Using $\alpha = \beta = 0$: $(\star, \star\star, \ast, \ast\ast) = (n, n, n+1, n)$ is used in \cite{Basting2017}, $(\star, \star\star, \ast, \ast \ast) = (n, n, n, n)$ in \cite{Murea2016} and $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n+1, n+1)$ in \cite{Langer2016}, and   $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n, n+1)$ in \cite{Landajuela2016}. 
\item Using $\alpha = \beta = 1$: $(\star, \star\star, \ast, \ast \ast) = (n+1, n, n, n+1)$ in \cite{Lozovskiy2018}, $(\star, \star\star,\ast, \ast\ast) = (n+1, n, n, n)$  in \cite{smaldone2014} and $(\star, \star\star, \ast, \ast\ast) = (n+1, n, n+1, n+1)$ in \cite{LeTallec2001, Wang2020}.
\end{itemize}


\begin{proposition}
\label{prop:monolithic_schemes}
By assuming well-posed, orientation-preserving deformation mappings, i.e. $(J^n)_{n \in \mathbb{N}}$ bounded in $L^{\infty}(\Omega^0)$, $J^n > 0$ for each $n \geq 0$, Problem \eqref{eq:discretized_monolithic_formulation} has unique solution for inf-sup stable finite element spaces if $\left( 2J^{\star\star} + J^{n+1} - J^{n} \right) > 0$ and $\alpha = \beta = 1$.
\end{proposition}

\begin{proof}
%The matrix system \eqref{eq:matrix_formulation} is solvable whenever $A_n^{-1}$ exists and $BA^{-1}_nB^{T}$ is invertible. 
Since all operators are bounded and inf-sup stable elements are used for velocity and pressure, it is enough to ensure that the bilinear form $\mathcal{A}$ is coercive.
%Thus its enough to ensure $\text{rank}(B) = m$ and $A_n$ positive definite.
%As standard in literature, let us evaluate \eqref{eq:lhs_bilinear_form_A} in $\mathbf{v} = \mathbf{u}$ and $q = p$. Integrating by parts the convective term and joining expressions, the following equality holds:
Indeed:
\begin{equation}
\label{eq:proof_monolithic_case}
\begin{aligned}
\mathcal{A}(\mathbf{u}, \mathbf{u}) = & \int_{\Omega^0} \frac{J^{\star}}{2\tau} \left( \frac{2 J^{\star\star}}{J_{\star}} + \alpha \frac{J^{n+1} - J^{n}}{J^{\star}} \right) \vert \mathbf{u} \vert^2  + % \int_{\Omega^0} 
J^{\star} 2\mu \vert \epsilon^{\star}(\mathbf{u}) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2} Div\bigg( J^{\star} H^{\star} \big( (\beta -1) \mathbf{u}^{\ast} - (\alpha-1) \mathbf{w}^{\ast\ast} \big) \bigg) \vert \mathbf{u} \vert^2 \, \text{d}\mathbf{X} 
\end{aligned}
\end{equation}
being the last quantity strictly positive under the stated assumptions.
\end{proof}


\begin{remark}
The extension of Proposition \ref{prop:monolithic_schemes} to the case with non-homogeneous Dirichlet boundary conditions follows from the trace theorem  by assuming $\Omega^0$ a Lipschitz bounded open set \cite{Ern2004}.
\end{remark}


\begin{corollary}
Assuming $\alpha = \beta = 1$, Problem \eqref{eq:discretized_monolithic_formulation} is well posed when:
\begin{itemize}
    \item  $3J^{n+1} - J^{n} > 0$ if $\star \star = n+1$, i.e. a restriction on the time step size.
    \item $J^{n+1} + J^{n} > 0$ if $\star \star = n$,  i.e. no restriction on the time step size, since we assume orientation preserving deformation mappings.
\end{itemize}
No restrictions apply to $\star,\ast, \ast\ast$.
\end{corollary}


\begin{proposition}
\label{prop:energy_estimate_monolithic}
Under assumptions of Proposition \ref{prop:monolithic_schemes} and $\alpha = \beta = 1, \star\star = n$, the scheme \eqref{eq:discretized_monolithic_formulation} is unconditionally energy stable with energy estimate:
\begin{equation}
\label{eq:energy_estimate_monolithic}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \mathbf{u}^{n+1} \vert^2 \, \text{d} \mathbf{X}  - \int_{\Omega^0} \rho \frac{J^n}{2\tau} \vert \mathbf{u}^n \vert^2 \, \text{d}\mathbf{X}
= -\int_{\Omega^0} 2\mu J^{\star} \vert \epsilon^{\star}(\mathbf{u}^{n+1}) \vert^2 \, \text{d} \mathbf{X} - \int_{\Omega^0} \frac{\rho}{2\tau} J^{n} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 \, \text{d} \mathbf{X}.
\end{equation}

\end{proposition}

\begin{proof}
By setting $\mathbf{v} = \mathbf{u}^{n+1}$ in the bi-linear form \eqref{eq:lhs_bilinear_form_A}, $q = p^{n+1}$ in forms \eqref{eq:remaining_forms} and manipulating terms as standard in literature, the energy equality follows:
\begin{equation}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X}  - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \mathbf{u}^n \vert^2 \, \text{d}\mathbf{X} = & \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{\star\star}) \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} + 
\int_{\Omega^0} \frac{\rho}{2\tau} (J^{\star\star} - J^n) \vert \mathbf{u}^n \vert^2 \, \text{d}\mathbf{X} & \\
& - \int_{\Omega^0} 2\mu J^{\star} \vert \epsilon^{\star} (\mathbf{u}^{n+1}) \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \frac{\rho}{2\tau} J^{\star\star} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2} Div(J^{\star} H^{\star} (\mathbf{u}^{\ast} - \mathbf{w}^{\ast\ast})) \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2} \alpha \frac{J^{n+1} - J^{n}}{\tau} \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2} Div\left( J^{\star} H^{\star} (\beta \mathbf{u}^{\ast} - \alpha \mathbf{w}^{\ast\ast}) \right) \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
Thus, for $\alpha=\beta=1$ and $\star\star = n$  the result follows.
\end{proof}


\begin{remark}
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 \cite{Formaggia2004} for the advection-diffusion problem for Crank-Nicolson (CN) and BDF(2). 
Therefore, we do not analyze here the schemes used in \cite{ failer2020impact, Hessenthaler2017,Tallec2003} -- based on CN and used in the context of fluid-solid interaction -- since their analysis repeats from \cite{Formaggia2004}. 
Also in the same context, some authors have used the generalized $\alpha$-methods since it is a popular scheme for elastodynamics  \cite{Liu2018}. 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.

\end{remark}


\section{Chorin-Temam schemes}
\label{sec:chorin_temam_schemes}


In the following, we describe a family of Chorin-Temam (CT) schemes for the iNSE-ALE problem, as we did for the monolithic case. %Such description keeps the freedom of choice for certain coefficients, which must be restricted to ensure unconditional stability.
Given $\mathbf{\widetilde V} $  a conforming space of $\mathbf{H}^1_0 (\Omega^0)$ and  $ \widetilde Q$ a conforming space of $ L^2_0 (\Omega^0) \cap H^1(\Omega^0)$, $\tilde{\mathbf{u}}^{0} \in \mathbf{\widetilde V}$, for $n\geq0$:% the proposed two-step CT schemes reads
\begin{enumerate}

    \item \textbf{Pressure-Projection Step $(\text{PPS})_{n}$}
    %Seeks a pressure term allowing the projection of $\tilde{\mathbf{u}}^{n+1}$ in the space of divergence-free solutions, in the form:
    Find $p^{n} \in \widetilde{Q}$ s.t.
    \begin{equation}
    \label{eq:chorin_temam_pps}
    \int_{\Omega^0} \frac{\tau}{\rho} J^{\circ} Grad(p^{n}) H^{\circ} : Grad(q) H^{\circ} \, \text{d}\mathbf{X} = -  \int_{\Omega^0} Div\left( J^{\circ} H^{\circ} \tilde{\mathbf{u}}^{n}\right) q \, \text{d}\mathbf{X} \quad \forall q \in \widetilde{Q}
    \end{equation}

    \item \textbf{Fluid-Viscous Step $(\text{FVS})_{n+1}$} %Seeks a tentative velocity  field $\tilde{\mathbf{u}}^{n+1}$ with pressure given explicitly: 
    Find $\tilde{\mathbf{u}}^{n+1} \in \mathbf{\widetilde{V}}$ s.t.
    \begin{equation}
    \label{eq:chorin_temam_fvs}
    \begin{aligned}
    \int_{\Omega^0} \rho J^{\star\star} \frac{\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^n}{\tau} \cdot \mathbf{v} \, \text{d}\mathbf{X} 
    + \int_{\Omega^0} \rho J^{\star} Grad(\tilde{\mathbf{u}}^{n+1}) H^{\star} (\tilde{\mathbf{u}}^{n} - \mathbf{w}^{\ast\ast}) \cdot \mathbf{v} \, \text{d}\mathbf{X} & \\
    + \int_{\Omega^0} J^{\star} 2 \mu \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) : \epsilon^{\star} (\mathbf{v}) \, \text{d}\mathbf{X} 
    - \int_{\Omega^0} Div(J^{\circ \circ} H^{\circ \circ} \mathbf{v}) p^n \, \text{d}\mathbf{X} & \\
    + \int_{\Omega^0} \frac{\rho}{2} \frac{J^{n+1} - J^{n}}{\tau} \tilde{\mathbf{u}}^{n+1} \cdot \mathbf{v} \, \text{d}\mathbf{X}
    + \int_{\Omega^0} \frac{\rho}{2} Div\left( J^{\star} H^{\star}(\tilde{\mathbf{u}}^{n} - \mathbf{w}^{\ast\ast})\right) \tilde{\mathbf{u}}^{n+1} \cdot \mathbf{v} \, \text{d}\mathbf{X} & = 0 \quad \forall \mathbf{v} \in \mathbf{\widetilde{V}}
    \end{aligned}
    \end{equation}
    
%    In particular, the corrected velocity is obtained through the update:
%    \begin{equation}
%    \begin{aligned}
%    \mathbf{u}^{n+1} & = \tilde{\mathbf{u}}^{n+1} - \frac{\tau}{\rho} Grad(p^{n+1}) F^{-1}_{\circ + 1} \text{ in } \Omega_0
%    \end{aligned}
%    \end{equation}
\end{enumerate}


The following energy estimate can be obtained under suitable conditions:
\begin{proposition}
\label{prop:energy_estimate_chorin_temam}
Under assumptions $\circ = \circ\circ = \star \star = n$, the solution to scheme \eqref{eq:chorin_temam_pps}-\eqref{eq:chorin_temam_fvs} is unconditionally stable, i.e.
%is unconditionally energy stable if $\circ = \circ \circ = \star\star = n$, without condition over $\star$. Moreover, the energy estimate is given by
\begin{equation}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau}  \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d} \mathbf{X} \leq & - \int_{\Omega^0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \, \text{d}\mathbf{X}  - \int_{\Omega^0} J^{n} \frac{\tau}{2\rho} \vert Grad(p^n) H^{n} \vert^2 \, \text{d}\mathbf{X} .
\end{aligned}
\end{equation}
\end{proposition}
\begin{proof}
As standard in literature, let us take $\mathbf{v} = \tilde{\mathbf{u}}^{n+1}$ in $(\text{FVS})_{n+1}$, and $q = p^n$ in $(\text{PPS})_{n}$. Adding both equalities and rewriting expressions, it follows:
\begin{equation}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} = & \int_{\Omega^0} \frac{\rho}{2\tau}(J^{n+1} - J^{\star\star}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2  \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \frac{\rho}{2\tau}(J^{\star\star} - J^{n}) \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} J^{\star\star} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} Div\left( J^{\circ\circ} H^{\circ\circ} (\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n}) \right) p^n \, \text{d}\mathbf{X} \\ 
& + \int_{\Omega^0} Div\left( (J^{\circ\circ} H^{\circ\circ} - J^{\circ} H^{\circ}) \tilde{\mathbf{u}}^n \right) p^n \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\tau}{\rho} J^{\circ} \vert (H^{\circ})^T Grad(p^n) \vert^2  \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{n}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}

Bounding the first divergence term using integration by parts and  Cauchy-Schwarz  inequality, it follows
\begin{equation}
\begin{aligned}
\int_{\Omega^0} Div\left( J^{\circ\circ} H^{\circ\circ} (\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n}) \right) p^n \, \text{d}\mathbf{X} & \leq \int_{\Omega^0} \frac{\rho}{2\tau} J^{\circ\circ} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2  \, \text{d}\mathbf{X} + \int_{\Omega^0} \frac{\tau}{2\rho} J^{\circ\circ}  \vert (H^{\circ\circ})^T Grad(p^n)  \vert^2 \, \text{d}\mathbf{x}
\end{aligned}
\end{equation}

Thus, the following energy estimate can be obtained:

\begin{equation}
\label{eq:energy_estimate_proof}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \leq & \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{\star\star}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} 
+ \int_{\Omega^0} \frac{\rho}{2\tau} (J^{\star\star} - J^{n}) \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} J^{\star\star} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2\tau} J^{\circ\circ} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\tau}{2\rho} J^{\circ\circ} \vert (H^{\circ\circ})^{T} Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} Div\left( (J^{\circ\circ} H^{\circ\circ} - J^{\circ} H^{\circ}) \tilde{\mathbf{u}}^{n}\right) p^n \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\tau}{\rho} J^{\circ} \vert (H^{\circ})^{T} Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{n}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}

From estimate \eqref{eq:energy_estimate_proof} it follows that whenever $\circ = \circ \circ = \star \star = n$ unconditional energy stability is attained, where $\star$ remains free of choice.
\end{proof}


\section{Numerical examples}
\label{sec:numerical_examples}


We consider a rectangular domain with opposite vertices $\{ (0, -1), (6, 1) \}$ where the iNSE-ALE formulation \eqref{eq:continuous_formulation} will be simulated over the interval $(0,\, 2) \, [s]$ with non-zero initial condition of the form $ \mathbf{u}(0) := \big(\gamma (1 - \mathbf{X}_1^2) \mathbf{X}_0 (6 - \mathbf{X}_0), 0\big) ,\, \gamma = 0.001$. 
The domain is deformed using $\mathcal{X}(\mathbf{X}, t) := \big( (1 + 0.9 sin(8 \pi t)) \mathbf{X}_0,\, \mathbf{X}_1 \big)$. %, i.e. an oscillation with initial expansion and frequency of $4 \pi$.

Discretization setup for Formulation \eqref{eq:discretized_monolithic_formulation} and \eqref{eq:chorin_temam_pps}-\eqref{eq:chorin_temam_fvs} is done choosing a time step $\tau = 0.01$ and space triangulation with elements diameter $h \approx 0.01 $, implemented through FEniCS \cite{FEniCS2015} using Python for interface and postprocessing.

To exemplify the theoretical results from previous sections, four schemes are taken into account. Monolithic (M) Formulation \eqref{eq:discretized_monolithic_formulation} is taken with linearized convective term and implicit treatment, i.e., $(\star, \ast, \ast\ast) = (n+1, n, n+1)$ where for $\star\star$ we consider two choices, denoted $\text{M}\, \star\star = n$ and $\text{M}\, \star\star = n+1$. 
For both cases the space discretization is carried out with $ \mathbf{V}/Q = [\mathbb{P}_2]^d/\mathbb{P}_1$ Lagrange finite elements. Similarly, Chorin-Temam (CT) scheme \eqref{eq:chorin_temam_fvs}-\eqref{eq:chorin_temam_pps} is taken with linearized convective term and implicit treatment, i.e. $(\star, \ast\ast, \circ, \circ\circ) = (n+1, n+1, n, n)$ with $\star\star$ as before, denoting each scheme by $\text{CT}\, \star\star = n$ and $\text{CT}\, \star\star = n+1$ with space discretization done through $\mathbf{\widetilde{V}}/\widetilde{Q} = [\mathbb{P}_1]^d/\mathbb{P}_1$ elements. 
In all cases homogeneous (equal to $\mathbf{0}$) boundary conditions are imposed for the velocity, zero-mean on the pressure and $\alpha=\beta=1$.

The results are assessed using time-dependent normalized parameters $ \hat{\delta}_{\text{M}}:= \delta_{\text{M}}/E_{st}^{\star}, \hat{\delta}_{\text{CT}}:=  \delta_{\text{CT}}/E_{st}^{\star}$ defined as: %, where $E_{st}^{\star}$ denote the \CB{\mod{strain energy}{viscous dissipation}} and $\delta_{M}, \delta_{CT}$ residual errors defined as:
\begin{equation}
\label{eq:energy_error}
\begin{aligned}
\delta_{M}^{n+1} &:= D^{n+1} + E_{st}^{\star} + \int_{\Omega_0} \frac{\rho J^{\star\star}}{2\tau} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 \, \text{d}\mathbf{X} , \quad \delta_{CT}^{n+1} := D^{n+1} + E^{\star}_{st} + \int_{\Omega_0} \frac{\tau J^{\circ}}{2 \rho} \vert (H^{\circ})^T Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
D^{n+1} &:= \int_{\Omega_0} \frac{\rho}{2\tau} \left( J^{n+1} \vert \mathbf{u}^{n+1} \vert^2 - J^n \vert \mathbf{u}^n \vert^2 \right) \, \text{d}\mathbf{X} , \quad E^{\star}_{st} = \int_{\Omega_0} 2 \mu J^{\star} \vert \epsilon^{\star} (\mathbf{u}^{n+1}) \vert^2 \, \text{d}\mathbf{X}.
\end{aligned}
\end{equation}


Figure \ref{fig:delta_hat_oscs} shows $\hat{\delta}_{\text{M}}, \hat{\delta}_{\text{CT}}$ values for each tested scheme. Propositions \ref{prop:energy_estimate_monolithic} and \ref{prop:energy_estimate_chorin_temam} are confirmed since $\hat{\delta}_{\text{M}}=0$ and $\hat{\delta}_{\text{CT}} \leq 0$ if $\star\star = n$. For $\star \star = n+1$, 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. 
%Moreover the expected energy decay from Propositions \ref{prop:energy_estimate_monolithic}, \ref{prop:energy_estimate_chorin_temam} for the choice $\star \star = n$  in $\text{M, CT}$ schemes is obtained.


%\CB{Recall that $\hat{\delta}_{\text{M}}, \hat{\delta}_{\text{CT}}$ in \eqref{eq:energy_error} are taken in such form since on scheme $\text{M}\, \star\star = n$ it follows $\hat{\delta}_{MT} = 0$ and similarly on $\text{CT}\, \star\star = n$ it follows $\hat{\delta}_{\text{CT}} \sim 0$, defining the base case.}

\begin{figure}[!hbtp]
	\centering
	\includegraphics[width=\textwidth]{figs/Comparison_Delta_Hat_Value_GCL_True_solver_LU.png}
	\caption{Summary of the numerical experiment in terms of energy balance. Left: Monolithic residual error values $\hat{\delta}_{\text{M}}$; 
    Right: Chorin-Temam residual error values $\hat{\delta}_{\text{CT}}$. %Both cases are simulated with deformation mapping $\mathcal{A}$ defined previously and shown on the interval $[0, 1] \, [s]$.}
    }
	\label{fig:delta_hat_oscs}
\end{figure}


\section{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 $\alpha=\beta=1$ with $\star \star = n$ as studied in \cite{LeTallec2001, Lozovskiy2018, smaldone2014, Wang2020}.
To the best of the authors knowledge, the unconditionally stable Chorin-Temam scheme derived in this work has not been reported yet. %, and moreover the numerical experiments studied here validate the energy stable propositions.


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