\begin{algorithm}
\caption{Sequential coupling for one time-step}\label{alg:stag}

$\mathbf{u}_{mesh}$ = $\left(\mathbf{x}_{n} - \mathbf{x}_{n-1} \right) \frac{1}{\Delta t}$\\
$\mathbf{u}_{n+1}$ = \text{Navier-Stokes} $(\mathbf{u}_n, \phi_n, \mathbf{u}_{mesh})$\\
$\phi_{adv}$ = \text{level set} $(\mathbf{u}_{n+1}, \phi_n)$\\
$\mathbf{x}_{n+1}$, $\text{front}_{n+1}$ = \text{X-Mesh} $(\mathbf{x}_n, \phi_{adv}, \phi_n, \text{front}_n)$\\
$\phi_{n+1}$ = \text{Fastmarching} $(\mathbf{x}_{n+1}, \text{front}_{n+1})$

$t$ = $t+\Delta t$ \\

\end{algorithm}

\begin{algorithm}
\caption{Sequential coupling for one time-step}
$\mathbf{u}_{mesh}$ = $\left(\mathbf{x}_{n} - \mathbf{x}_{n-1} \right) \frac{1}{\Delta t}$\\
$\mathbf{u}_{n+1}$ = \text{Navier-Stokes} $(\mathbf{u}_n, \phi_n, \mathbf{u}_{mesh})$\\
$\phi_{adv}$ = \text{level set} $(\mathbf{u}_{n+1}, \phi_n)$\\
$\mathbf{x}_{n+1}$, $\text{front}_{n+1}$ = \text{X-Mesh} $(\mathbf{x}_n, \phi_{adv}, \phi_n, \text{front}_n)$\\
$\phi_{n+1}$ = \text{Fastmarching} $(\mathbf{x}_{n+1}, \text{front}_{n+1})$
$t$ = $t+\Delta t$ \\
\end{algorithm}
{
    "title": "x-mesh: a new approach for the simulation of two-phase flow with sharp   interface",
    "id": "2302.03983",
    "abstract": "accurate modeling of moving boundaries and interfaces is a difficulty present in many situations of computational mechanics. we use the extreme mesh deformation approach (x-mesh) to simulate the interaction between two immiscible flows using the finite element method, while maintaining an accurate and sharp description of the interface without remeshing. in this new approach, the mesh is locally deformed to conform to the interface at all times, which can result in degenerated elements. the surface tension between the two fluids is added by imposing the pressure jump condition at the interface, which, when combined with the x-mesh framework, allows us to have an exactly sharp interface. if a numerical scheme fails to properly balance surface tension and pressure gradients, it leads to numerical artefacts called spurious or parasitic currents. the method presented here is well balanced and reduces such currents down to the level of machine precision.",
    "categories": "cs.ce",
    "doi": "10.1016/j.jcp.2024.112775",
    "created": "2023-02-08",
    "updated": "2024-02-01",
    "authors": [
        "antoine quiriny",
        "jonathan lambrechts",
        "nicolas mo\u00ebs",
        "jean-fran\u00e7ois remacle"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2302.03983"
}
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