## 2 Mathematical Models and Methods
### Gas-Water Interface Treatment  
As mentioned above, we adopted the RGFM method proposed by [PERSON] et al. [34] to deal with the conditions at the gas-water interface. For the details of the RGFM method, one can refer to [34]. Here, the application of this method in the present study is only briefly described through the following steps:  
1. On one side of the interface, an adjacent point closest to the interface is selected. Then, a corresponding point on the other side of the interface is found that has a minimum dot product of the normal vectors. Then, a pair of Riemann points is constructed. 2. Use the parameter values (density, normal velocity, pressure) of the Riemann points on both sides of the interface to construct a Riemann problem. Then, an Approximate Riemann Problem Solver (ARPS) [39; 40] is used to predict the interface values such as the density and the pressure. This ARPS is developed based on a double-shock characteristics structure, which can achieve a second-order accuracy in solving the gas-water Riemann problem [39]. Then, the predicted interface values (pressure, normal velocity and entropy) are taken to replace the values of the adjacent point on the real fluid side. 3. The above processes (I) and (II) are repeated until all the adjacent points on one side of the interface are re-defined. Then, the values at these adjacent points are extrapolated to the other side of the interface where the \"ghost points\" are distributed, by iteratively solving the following extrapolation equation [26]: \[\frac{\partial I}{\partial t}\pm(n_{x}\frac{\partial I}{\partial x}+n_{y}\frac {\partial I}{\partial y}+n_{z}\frac{\partial I}{\partial z})=0.\] (7) In Equation (7), \(I\) represents the variables to be extrapolated such as entropy from the real fluid side to the ghost points. The unit normal vector of the interface, \(\vec{N}=(n_{x},n_{y},n_{z})\), is calculated as \(\vec{N}=\frac{\
abla\phi}{|\
abla\phi|}\). Here, \(\phi\) is the level-set function which is defined as a signed distance from a point in the flow field to the interface. When \(\phi\) equals 0, the position of interface is indicated. The direction of \(\vec{N}\) always points from the region with a negative value of \(\phi\) to the region with a positive \(\phi\)When the variables are extrapolated from a region of \(\phi<0\) to \(\phi\geq 0\), \"\(+\)\" is used in Equation (7). Otherwise, \"\(-\)\" is chosen. The ghost points are distributed in a domain that possesses a distance to the interface less than twice of the typical grid length according to the TVD scheme used in the present study. When the values of all the ghost points corresponding to the adjacent points on one side of the interface are given, the ghost points on the other side of interface are defined in the same way. 4. Advancement of the flow field for one time step for both sides of the interface, so that two sets of the solutions to the Euler equations are obtained. 5. Advancement of the location of the interface for one time step by solving the level-set equation: \[\frac{\partial\phi}{\partial t}+u\frac{\partial\phi}{\partial x}+v\frac{ \partial\phi}{\partial y}+w\frac{\partial\phi}{\partial z}=0.\] (8) This level-set method has proved to be capable of capturing the multi-medium interface successfully and also the interaction between a shock wave and the interface [33]. However, when the simulation proceeds, the level-set function \(\phi\) tends to be distorted, and thus cannot maintain the property as a signed distance function. Therefore, a re-initialization process is needed [41], which is performed after every time step in the present study. After locating the new position of the interface, we use the updated level-set function to determine which solution obtained in process (IV) should be chosen. 6. Proceed to the next time step. The level-set Equation (8) is discretized by a fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme [42] in the present study. For more details of this scheme, the reader is referred to [26; 42].