# Sharp Interface Capturing in Compressible Multi-Material Flows with a Diffuse Interface Method
## 3 Time and Space Discretization  
### Time Discretization  
The time segment \([0,T]\) is split into successive time-steps \([t^{n},t^{n+1}]\) with \(\Delta t\equiv\Delta t^{n}=t^{n+1}-t^{n}>0\) and \(n\in\mathbb{N}\). We further identify any variable or parameter \(\psi\) at time \(t^{n}\) as \(\psi^{n}\). In this study we first consider a semi-discrete, first-order, semi-implicit time discretization scheme, where discrete spatial operators are written in an explicit form, except in the pressure terms. The pressure terms are made implicit for numerical stability reasons and to make use of larger time steps, choice of which is restricted by CFL-like [29] conditions based on material velocities. This gives rise to a coupled, nonlinear system of semi-discrete equations:  
\[\rho_{\phi}^{n+1}\alpha_{\phi}^{n+1}=\rho_{\phi}^{n}\alpha_{\phi}^{n}-\Delta t \,\
abla\cdot(\rho_{\phi}^{n}\alpha_{\phi}^{n}\mathbf{u}_{\phi}^{n}), \tag{11}\]  
\[\rho_{\phi}^{n}\alpha_{\phi}^{n}\left(\mathbf{u}_{\phi}^{n+1}-\mathbf{u}_{\phi}^{n} \right)=-\Delta t\,\rho_{\phi}^{n}\alpha_{\phi}^{n}\mathbf{u}_{\phi}^{n}\cdot\
abla (\mathbf{u}_{\phi}^{n})-\Delta t\,\alpha_{\phi}^{n}\,\
abla\,\rho_{\phi}^{n+1}+ \Delta t\,\mathbf{F}_{\phi}^{n+1}, \tag{12}\]  
\[\rho_{\phi}^{n+1}\alpha_{\phi}^{n+1}\epsilon_{\phi}^{n+1}=\rho_{\phi}^{n}\alpha _{\phi}^{n}\epsilon_{\phi}^{n}-\Delta t\,\
abla\cdot(\rho_{\phi}^{n}\alpha_{ \phi}^{n}\epsilon_{\phi}^{n}\mathbf{u}_{\phi}^{n})-\Delta t\,r_{\phi}^{n+1}\left[ \frac{\alpha_{\phi}^{n+1}-\alpha_{\phi}^{n}}{\Delta t}+\
abla\cdot(\alpha_{ \phi}^{n}\mathbf{u}_{\phi}^{n})\right], \tag{13}\]  
\[p_{\phi}^{n+1}=p^{n+1}, \tag{14}\]  
\[\sum_{\phi}\alpha_{\phi}^{n+1}=1, \tag{15}\]  
\[\text{EOS}_{i}(\rho_{\phi}^{n+1},\epsilon_{\phi}^{n+1},T_{\phi}^{n+1},p_{\phi }^{n+1})=0,\quad i=1,2. \tag{16}\]  
Here superscripts \(n\) and \(n+1\) denote the term at previous discrete time \(t^{n}\) (explicit terms) and next discrete time \(t^{n+1}\) (implicit terms), respectively. The above system of partial differential Equations (11)-(16) is in the semi-implicit form in time. With regard to the time scheme, we have taken a two-step approach for the numerical solution of the system (11)-(16). These are:  
1. Explicit convection/advection step: This step deals with the numerical calculation of all explicit terms in the system (11)-(16). This step is also called a convection/advection step (referred to as advection step hereafter) because these explicit terms can be determined by simply solving the advection problems for \(\mathbf{u}_{\phi}\), \(\alpha_{\phi}\), \(\alpha_{\phi}\rho_{\phi}\) and \(\alpha_{\phi}\rho_{\phi}\epsilon_{\phi}\), as shown below:  
**Time discrete system for the explicit advection step** \[(\mathbf{u}_{\phi})^{c,n+1}=\mathbf{u}_{\phi}^{n}-\Delta t\,\mathbf{u}_{\phi}^{n} \cdot\
abla\mathbf{u}_{\phi}^{n},\] (17) \[(\alpha_{\phi})^{c,n+1}=\alpha_{\phi}^{n}-\Delta t\,\
abla\cdot( \alpha_{\phi}^{n}\mathbf{u}_{\phi}^{n}),\] (18) \[(\rho_{\phi}\alpha_{\phi})^{c,n+1}=\rho_{\phi}^{n}\alpha_{\phi}^{n} -\Delta t\,\
abla\cdot(\rho_{\phi}^{n}\alpha_{\phi}^{n}\mathbf{u}_{\phi}^{n}),\] (19) \[(\rho_{\phi}\alpha_{\phi}\epsilon_{\phi})^{c,n+1}=\rho_{\phi}^{n} \alpha_{\phi}^{n}\epsilon_{\phi}^{n}-\Delta t\,\
abla\cdot(\rho_{\phi}^{n} \alpha_{\phi}^{n}\epsilon_{\phi}^{n}\mathbf{u}_{\phi}^{n}).\] (20)  
where superscript \(c\) denotes the advected quantities. **Remark 1**. : _In Section 5 we will focus on the application of the sharp interface capturing strategy for the space discretization of divergence operators in Equations (18)-(20)._  
1. Implicit step: This step deals with the numerical solution of only implicit terms in the system (11)-(16) after calculating the explicit ones by solving the explicit advection step. To rewrite the system (11)-(16) in form of only implicit quantities, we perform the following substitution: (19) in (11); (17) in (12); (20) and (18) in (13).