# Sharp Interface Capturing in Compressible Multi-Material Flows with a Diffuse Interface Method
## 5 Modification of the Advection Step: Sharp Interface Capturing (IC) Approach  
Interface Capturing (IC) Strategy: Multidimensional Limiting Process with the Upper Bound Limiter (MLP-UB)  
To ensure the sharp interface capturing in the solution of the compressible multi-material model, the reconstruction of the scalars \(\Pi(\alpha_{\phi},\rho_{\phi},e_{\phi})\) in each finite volume cell during the advection step depicted in time algorithm (see Figure 3) is done using a second-order multidimensional MUSCL slope reconstruction strategy [19], where the predicted slope/gradient is limited using a recently developed, compressive, shape-preserving limiter, referred to as _Multidimensional Limiting Process-Upper Bound (MLP-UB)_ (see [18]), which is an updated version of the _MLP_ methodology detailed in [20]. We briefly elaborate this new MLP-UB method for a generic scalar variable \(z(\mathbf{x},t)\) advected in a medium with velocity \(\mathbf{u}\):  
\[\frac{\partial z}{\partial t}+\
abla\cdot(z\mathbf{u})=0. \tag{41}\]Let us use \(K\) as a generic notation for a given finite volume cell, \(A\) as an edge (or a face in 3D) of this volume \(K\), and a vector \(\
u_{A}\) being a normal unit vector to \(A\) pointing out of the cell \(K\) (see Figure 4). We look for the semi-discrete finite volume schemes of the form:  
\[\frac{dz_{K}}{dt}=-\frac{1}{|K|}\sum_{A\subset\partial K}|A|\,z_{A}\,\mathbf{u}_{A }\cdot\
u_{A}. \tag{42}\]  
where \(z_{K}\) is the value of \(z\) defined at the center of a cell \(K\), and \(z_{A}\) (resp. \(\mathbf{u}_{A}\)) denotes a certain value of \(z\) (resp. \(\mathbf{u}\)) at the midpoint \(\mathbf{x}_{A}\) of an edge \(A\). As usual for a finite volume scheme, flux is linearly reconstructed in each cell, and a slope limiter is applied to limit this reconstructed gradient. The general expression of a MUSCL [19] linear reconstruction of a discrete field \(z_{K}\) in cell \(K\) is:  
\[\mathcal{I}_{K^{2}}^{h}z(\mathbf{x})=z_{K}+\Theta_{K}\,(\
abla^{h}z)_{K}\cdot(\bm {x}-\mathbf{x}_{K}). \tag{43}\]  
where \(\mathbf{x}_{K}\) is the center of mass of cell \(K\), \((\
abla^{h}z)_{K}\) is a predicted estimate of the gradient, and \(\Theta_{K}\) is a positive scalar intended for limiting this gradient. Next the reconstructed value \(z_{A}\) at the midpoint \(\mathbf{x}_{A}\) of an edge \(A\) in cell \(K\) is given by:  
\[z_{A}=\mathcal{I}_{K}^{h}z(\mathbf{x}_{A})=z_{K}+\Theta_{K}\,(\
abla^{h}z)_{K} \cdot(\mathbf{x}_{A}-\mathbf{x}_{K}).\]  
which, in case of a Cartesian mesh with \(h\) as a constant distance between cell centers, can be simplified as:  
\[z_{A}=z_{K}+\frac{h}{2}\,\Theta_{K}\,(\
abla^{h}z)_{K}\cdot\
u_{A}.\]  
Here we summarize main steps of the MLP-UB method. Detailed analysis of these steps can be found in [18]. Step 1: Prediction of the gradient  
In this study, gradient is predicted with a 8-point formula (see [18]). This 8-point formula for calculating the gradient of a scalar \(z_{K}\) in cell \(K\) of a 2D Cartesian mesh (see Figure 5) is:  
\[(\
abla^{h}z)_{K}=\frac{1}{h}\left( \begin{array}{c}\frac{1}{12}(z_{K_{8}}-z_ {K_{6}})+\frac{1}{3}(z_{K_{5}}-z_{K_{4}})\\ +\frac{1}{12}(z_{K_{3}}-z_{K_{1}})\\ \frac{1}{12}(z_{K_{8}}-z_{K_{3}})+\frac{1}{3}(z_{K_{7}}-z_{K_{2}})\\ +\frac{1}{12}(z_{K_{6}}-z_{K_{1}})\end{array} \right). \tag{44}\]  
Figure 4: Geometric entities of a finite volume cell \(K\): mass center \(x_{K}\), edge \(A\), edge center \(\mathbf{x}_{A}\), normal unit vector \(\
u_{A}\), corner \(c\). In order to define the scalar limiting functions \(\Theta_{c}\) for each corner of a cell \(K\), we compute a _non-limited_ (NL) reconstructed value with the predicted centered gradient:  
\[\mathcal{I}_{K}^{NL}z(\mathbf{x})=z_{K}+(\
abla^{h}z)_{K}\cdot(\mathbf{x}-\mathbf{x}_{K}).\]  
We denote it by \(\hat{z}_{K}(\mathbf{x})=\mathcal{I}_{K}^{NL}z(\mathbf{x})\) for convenience in the next formulae, and we define the extremum value \(z_{c}^{extr}\) as:  
\[z_{c}^{extr}=\left\{