## Section: New Results

### Modeling and numerical simulation of complex fluids

In [25], Giacomo Dimarco, RaphaĆ«l Loub ere, Jacek Narski, and Thomas Rey extend the Fast Kinetic Scheme (FKS) originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with conservative fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic.

In the context of the PhD of Claire Colin-Lecerf, C. Calgaro and co-authors derive in [45] a combined Finite Volumes - Finite Elements (CFVFE) scheme. This work can be seen as a generalization of some previous contributions on incompressible flows [5], [4], [6], in the context of a low-Mach model. Here, the temperature obeying an energy law has been taken into account. The authors chose to solve the continuity equation and the state equation linking temperature, density and thermodynamic pressure is imposed implicitly. Now the velocity field is no more divergence-free, so that the projection method solving the momentum equation has to be adapted. This combined scheme preserve the constant state and ensure the discrete maximum principle on the density. Their numerical results have been compared to some others which use purely finite elements schemes (see [62], [58], [81]) and in particular on a benchmark consisting in a transient hot jet entering in a cavity.

Diffuse interface models, such as the Kazhikhov-Smagulov model, allow to describe some phase transition phenomena. The theoretical analysis of this model was given by Bresch at al. [64] (see also reference therein). In the previous work [6], C. Calgaro et al. have implemented the CFVFE scheme and studied numerically the progression of the front of a powder-snow avalanche with respect to some characteristics parameters of the flow, such as the Froude, Schmidt and Reynolds numbers. In [18], C. Calgaro and co-authors investigate theoretically the CFVFE scheme. They construct a fully discrete numerical scheme for approximating the two-dimensional Kazhikhov-Smagulov model, using a first-order time discretization and a splitting in time to allow the construction of the combined scheme. Consequently, at each time step, one only needs to solve two decoupled problems, the first one for the density (using the Finite Volume method) and the second one for the velocity and pressure (using the Finite Element method). The authors prove the stability of the combined scheme and the convergence towards the global in time weak solution of the model. In this model, the convection-diffusion equation for the density can also be discretized by a implicit-explicit (IMEX) second order method in the Finite Volume scheme. In the framework of MUSCL methods, C. Calgaro and M. Ezzoug prove in [36] that the local maximum property is guaranteed under an explicit Courant-Friedrichs-Levy condition and the classical hypothesis for the triangulation of the domain.