## 1 Introduction
The propagation of the underwater shock waves and the stress exerted on the boundary caused by the collapse of bubbles were also visualized [13; 14]. Since the year 2000, experiments of UNDEXs were conducted with the aim of investigating the interaction between the shock waves and the structures nearby [3; 15; 16]. The pressure pulses associated with the temporal evolution of bubbles during the small-charge underwater explosion were also captured by [PERSON] et al. [17] by using high-speed camera and pressure sensors. However, these kinds of experiments are relatively costly and hazardous. In addition, the data acquired from the experiments is limited due to the restriction of the present observational technique. However, since the 1950s, numerical methods have developed vigorously and become an efficient tool for the investigation of UNDEXs. The earlier numerical studies of the phenomenon [18; 19] were undertaken focusing on the propagation and decay of the shock wave in the water. In recent decades, due to the rapid advancement of the computational resources and the simulation techniques, numerous modeling work of UNDEXs were made. To name a few, [PERSON] [20] used the Finite Element Analysis (FEA) codes to model the important physical processes related to UNDEXs. Their major characteristics such as the initiation of the explosion and the transmission of the shock wave were successfully captured in their simulations. In the year 1998, [PERSON] et al. [21] applied a two-dimensional model to calculate the response of a surface ship structure to the underwater shock as well as the following cavitation caused by the collapse of bubbles. A good quantitative agreement between their computational results and the literature was also found. Later, the Smoothed Particle Hydrodynamics (SPH) method was applied in simulatingthe UNDEX problems [22; 23; 24], and a quantitative measurement of the strength and the location of the shock wave during the explosion was obtained in these studies. A recent three-dimensional modeling of the UNDEX was performed by [PERSON] et al. [25], with the aim of simulating a shock-induced collapse of the gas bubbles in the water by using a finite-volume interface capturing method. However, although there exists a large quantity of numerical studies of UNDEXs as discussed above, problems still occur in the simulations when a large deformation of the gas-water interface happens. Moreover, errors are introduced due to the smearing at the interface, which may cause non-physical oscillation at the multi-medium interface if no special treatment is made. The inaccuracy of UNDEX simulations is also greatly attributed to the difficulty of capturing the movement of the gas-water interface with a high pressure ratio. At present, two types of the numerical methods are usually adopted to capture the interface, namely front capturing method and front tracking method. In the front capturing method, the discontinuity at the interface is approximated as a continuous solution with a steep gradient over a few grid nodes. This method is simple to implement and also suitable for dealing with large deformation of the interface. However, due to the smearing of the fluid properties, such as its density and viscosity, in the vicinity of the interface, non-physical oscillations occur. Moreover, the difference of the properties on both sides of the interface also causes numerical errors. In contrast to that, in the front tracking method, the interface is treated as a moving internal boundary. Further treatments such as the distribution of virtual grid nodes on both sides of the interface are also made. This type of method does not smear out the discontinuity of the properties when dealing with the interface problem, and is able to provide satisfactory results. However, the disadvantage of this method is that it is unsuitable for dealing with problems with a large interface deformation. In addition, the extension to multi-dimensional simulations is relatively complicated.