# Sharp Interface Capturing in Compressible Multi-Material Flows with a Diffuse Interface Method
## 8 Conclusions and Scope for the Future Work  
In the present work, we have considered the hyperbolic compressible Euler system of equations, which can serve as the basis for numerical simulations of the several industrial applications in the domain of the compressible multi-material flows. We have worked with this model in the limit of a full mechanical equilibrium, and also considered an instantaneous pressure equilibrium. Within the scope of such a mathematical model, we focused on the problem of the interface sharpening, where we chose to work with the Interface Capturing approach. We performed the analysis of interface sharpening using a recently developed Diffuse Interface Method: Multidimensional Limiting Process-Upper Bound (MLP-UB) [18]. This method has been proposed in a recent paper [18], where the authors have demonstrated its robustness for the pure advection cases. The present paper is thus considered as a natural sequel to the work presented in [18].  
For performing the numerical simulations in this work we have used our in-house, high-performance, modern software: MMICADO; within its framework, we have first proposed a robust time scheme, where we first explicitly calculate the advection/convection terms. Within the framework of Finite Volume space discretization on 2D Cartesian meshes, the divergence terms in these advection/convection equations are computed by employing a second order accurate, multidimensional MUSCL [19] slope reconstruction strategy along with the MLP-UB gradient limiting method. A 1D analysis is also presented to highlight the choice made for scalar reconstruction in their primitive form (instead of their conservative form) by using the most and the least compressive form of MUSCL+MLP-UB approach.  
Finally, we have provided three complex multi-material test cases, where our aim was to produce accurate results on the coarse 2D Cartesian meshes. Looking at the results we can conclude that our approach has produced:  
1. Accurate and sharp capturing of the intricate interface evolution without numerical artifacts.
2. Accurate interaction of the shocks with the multi-material interfaces.
3. Bounded numerical results for any number of materials.  
A direct extension of the work presented in this paper is to take into account the phenomena of phase change by considering the heat and mass transfer across the multi-material interfaces. Another work which is conceptualized is the study of _sliding interfaces_, where we want to consider the relative motion between the multi-material interfaces. Another immediate work concerns the verification and validation of the 3D version, and theoretical demonstration of the positivity preserving property (boundedness) of the method. We plan to address these issues in the future research articles.  
Conceptualization, C.F., M.P. and R.M.; Data curation, S.N.; Formal analysis, S.N.; Investigation, S.N.; Analysis, S.N., C.F., M.P. and F.D.V.; Methodology, C.F., S.N. and M.P.; Project administration, C.F., M.P. and R.M.; Resources, C.F. and M.P.; Software, S.N., C.F. and M.P.; Supervision, C.F. and M.P.; Validation, S.N.; Visualization, S.N.; Writing original draft, S.N.; Writing review and editing, S.N., M.P. and F.D.V. All authors have read and agreed to the published version of the manuscript.  
No external funding reported.  
Not applicable.  
The present paper contains all the necessary data needed for the thorough understanding of the work. In case a requirement for any supplementary material arises, the readers are encouraged to contact the corresponding authors.  
The authors declare no conflict of interest.