# Sharp Interface Capturing in Compressible Multi-Material Flows with a Diffuse Interface Method
## 1 Introduction  
Multi-phase compressible flows are present in a wide and ever expanding range of natural and industrial applications. Among them we can mention supernova explosions [1], high speed flows in jet and rocket propulsion [2], underwater explosions [3, 4], and the scenario of vapor explosions in post accidental situations in nuclear reactors [5, 6]. To compute these flows, two main families of numerical methods are used: Lagrangian and Eulerian. In the Lagrangian approach, the objective is to describe the phenomena that govern the state of an individual fluid element, while in the Eulerian approach, the objective is to describe the phenomena that govern the state of the fluid in a fixed volume element. In these modeling approaches, the main concerns are the treatment of the interfaces and the phenomena associated with them. In the case of a Lagrangian approach, the interfaces are followed by moving and deforming the mesh [7, 8]. For the Eulerian approach, it is possible to capture or reconstruct the interface by adding a dedicated equation. This is the approach adopted in the methods such as Volume Of Fluid (VOF) [9, 10, 11] and Level Set [12, 13]. Front tracking methods [14, 15] also exist, coupling some advantages of the Lagrangian and Eulerian representations. These methods consider the interface as a local discontinuity. However, algorithms dedicated for the determination of the position of interfaces are often expensive. Moreover, strong topology changes and formation/disappearance of the interfaces are particularly complex to deal with, especially with the increase in the number of phases present. There is another class of Eulerian methods called _algebraic_ methods, which belong to the class of sharp Interface Capturing (IC) approach [16]. These methods are often referred to as _Diffuse Interface Methods (DIM)_[17], an exhaustive up-to-date review of which can be found in [18]. In this family of methods, the diffusion of the interface is allowed. The interfaces are not subjected to any particular geometrical treatment, which makes their implementation easier. Moreover, these methods can handle the dynamic formation, disappearance and large deformation of the interfaces in flows with several phases. The propagation of the acoustic waves can also be naturally taken into account. However, one of the drawbacks of these methods is the smearing of the already diffused interfaces, due to the numerical diffusion. Significant smearing of the interfaces can then result in an extended zone of a pure numerical (non-physical) mixture of the different materials in the vicinity of the interfaces. Thus, the accurate description of any physical phenomena at the interface may be deteriorated by this wide diffusion zone. For our own end-use applications, we have decided to focus in this paper on the IC approach to solve the multi-material compressible problem. To ensure the sharp interface capturing in the solution of a compressible multi-phase model, a second-order multidimensional MUSCL [19] reconstruction is used with a recently developed compressive, shape-preserving gradient limiter, referred to as _Multidimensional Limiting Process-Upper Bound (MLP-UB)_[18], which is an updated variant of the classical _MLP_ approach demonstrated in [20]. Sharp interface capturing methods are generally Finite Volume (FV) methods, either based on the high order flux evaluations or on the slope reconstruction, but in all the cases they make use of the limiters (flux or slope). Besides the MLP-UB approach adopted here, other popular methods/schemes found in the literature are: limited downwind scheme [21], THINC scheme [22] or the method using correction algorithms [23]. The one-pressure, compressible system of the continuous equations considered in this work is focused on the phase volume fractions together with the evolution of the mean mass, momentum and energy of each phase [24; 25; 26].