Compressible multiphase flows appear in many applications, from aerospace and environmental engineering to biomechanics. Over the last decades, several mathematical models have been proposed to describe the interaction of different phases. Among the most widely used are the Baer–Nunziato model [1], the Kapila model [2], and more recent thermodynamically compatible formulations such as those proposed by Romenski et al. [3]. These models have significantly advanced the simulation of compressible mixtures.

Despite these advances, important challenges remain. Most existing models are designed for specific situations and do not provide a unified framework able to account simultaneously for surface tension, phase transition, and elastic effects. In particular, including capillarity often leads to higher-order or dispersive terms, which complicate both the mathematical analysis and the numerical approximation [4], [5]. In addition, coupling multiphase flows with deformable elastic materials is still largely an open problem [6].

On the numerical side, significant progress has been made in the last years in the development of high-order accurate numerical methods, as well as in structure-preserving or thermodynamically compatible schemes designed to maintain key physical and mathematical properties of the continuous model at the discrete level [7], [8]. However, combining both in a single numerical framework remains highly challenging. Efficient and robust high-order structure-preserving methods on unstructured meshes are still under active research [9].

The goal of COPERNICUS is to contribute to multiphase flows from both the theoretical and numerical points of view by developing a new unified hyperbolic and thermodynamically compatible framework for compressible flows with capillarity, extendable to elastic media, together with accurate and efficient high-order structure-preserving numerical methods.

1. M.R. Baer, J.W. Nunziato.
   A two-phase mixture theory for the deflagration-to-detonation transition in reactive granular materials.
   International Journal of Multiphase Flow, Vol. 12(6), pp. 861–889, 1986.
2. A.K. Kapila, R. Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart.
   Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations.
   Physics of Fluids, Vol. 13(10), pp. 3002–3024, 2001.
3. E. Romenski, D. Drikakis, E.F. Toro.
   Conservative models and numerical methods for compressible two-phase flow.
   Journal of Scientific Computing, Vol. 42(1), pp. 68–95, 2010.
4. S. Gavrilyuk.
   Multiphase Flow Modeling via Hamilton’s Principle.
   In Variational Models and Methods in Solid and Fluid Mechanics, CISM Courses and Lectures, Vol. 535, pp. 163–210, 2011.
5. S. Gavrilyuk, R. Saurel.
   Mathematical and Numerical Modeling of Two-Phase Compressible Flows with Micro-Inertia.
   Journal of Computational Physics, Vol. 175(1), pp. 326–360, 2002.
6. S. Gavrilyuk, N. Favrie, R. Saurel.
   Modelling wave dynamics of compressible elastic materials.
   Journal of Computational Physics, Vol. 227(5), pp. 2941–2969, 2008.
7. S. Busto, M. Dumbser, I. Peshkov, E. Romenski.
   On thermodynamically compatible finite volume schemes for continuum mechanics.
   SIAM Journal on Scientific Computing, Vol. 44(3), pp. A1723–A1751, 2022.
8. R. Abgrall, S. Busto, M. Dumbser.
   A simple and general framework for the construction of thermodynamically compatible schemes for computational fluid and solid mechanics.
   Applied Mathematics and Computation, Vol. 440, 127629, 2023.
9. R. Abgrall, M. Ricchiuto.
   High-order methods for CFD.
   Encyclopedia of Computational Mechanics, 2017.