## Section: New Results

### Mathematical modeling and numerical methods for Partial Differential Equations

#### Mathematical analysis

Participants : Céline Grandmont, Muriel Boulakia.

In the submitted paper [43] , the coupling between a compressible fluid and an elastic structure is studied. M. Boulakia and S. Guerrero establish the local in time existence and the uniqueness of regular solutions for this model. Contrary to most of the works on this subject, the equations do not contain extra regularizing terms. The result is proved by first introducing a linearized problem and by proving that it admits a unique regular solution. The regularity is obtained through successive estimates on the unknowns and their derivatives in time and through elliptic estimates. At last, a fixed point theorem allows to prove the existence and uniqueness of a regular solution of the nonlinear problem.

**Work in progress**

M. Boulakia and S. Guerrero are also working on the controllability of fluid-structure interaction problems for a rigid structure immersed in an incompressible fluid. The control acts on a small subdomain of the fluid domain. This work generalizes the paper [58] and deals with the three-dimensional case without any restriction on the geometry of the solid.

#### Numerical methods in fluid dynamics

Participants : Matteo Astorino, Franz Chouly, Miguel Ángel Fernández, Céline Grandmont, Jean-Frédéric Gerbeau, Jimmy Mullaert.

This activity on fluid-structure interaction is done in close collaboration with the MACS project-team.

##### Algorithms for fluid-structure interaction problem

Following the results reported in [12] , M. Astorino, F. Chouly and M.A. Fernández have proposed a semi-implicit coupling scheme for the numerical simulation of fluid-structure interaction systems involving a viscous incompressible fluid. The scheme is stable irrespectively of the so-called added-mass effect and allows for conservative time-stepping within the structure. The efficiency of the scheme is based on the explicit splitting of the viscous effects and geometrical/convective non-linearities, through the use of the Chorin-Temam projection scheme within the fluid. Stability comes from the implicit pressure-solid coupling and a specific Robin treatment of the explicit viscous-solid coupling, derived from Nitsche's method. These results and some numerical experiments have been reported in [13] .

M.A. Fernández has shown that the stabilized explicit coupling scheme reported in [20] can be cast into a Robin-Robin coupling framework. In particular, this allows the introduction of a general class (i.e. not necessarily within the Nitsche framework) of stabilized explicit coupling schemes based on genuine Robin-Robin transmission conditions. Interestingly, if a Chorin-Temam scheme is used in the fluid, this Robin-Robin interface treatment provides natural stabilization of explicit coupling. The convergence behavior needs, however, further investigations. Some of these results and numerical simulations illustrating the properties of the proposed algorithms have been presented by M.A. Fernández at the 15th International Conference on Finite Elements in Flow Problems (FEF09), April 1–3, 2009, Tokyo, Japan.

In collaboration with C. Farhat, A. Rallu and K. Wang (Stanford), J.-F. Gerbeau studied fluid-structure interaction problems associated with underwater implosions [49] . The structures are supposed to be immersed within the fluid and their two respective grids are completely independent. A numerical method has been proposed to easily switch from a fluid-structure to a fluid-fluid configuration in order to handle the apparition of cracks in the structure.

J.-F. Gerbeau, in collaboration with C. Farhat and A. Rallu (Stanford), proposed an original method to efficiently solve exact Riemann problems of Gas dynamics for arbitrary equations of state (in particular JWL). The algorithm relies on the construction of a metamodel based on sparsegrids [48] .

##### Parareal time-stepping

F. Chouly and M.A. Fernández have investigated a parallel time-marching scheme for coupled parabolic-hyperbolic problems, as a prototype of fluid-structure interaction problems involving a linear structure and a viscous fluid. No linearity assumption is made on the parabolic side. The standard Parareal scheme is applied to the parabolic part, while the modified algorithm proposed in [62] is applied to the hyperbolic part. This hybrid Parareal treatment relies on the partitioned formulation of the coupled propagator. Numerical evidence shows that the resulting scheme is stable for a wide range of physical and discretization parameters. This work has been reported in [27] .

##### Stabilized finite element

E. Burman (University of Sussex) and M.A. Fernández have proposed a new analysis for the stabilized PSPG (Pressure Stabilized Petrov Galerkin) method applied to the transient Stokes problem. Stability and convergence are obtained under different conditions on the discretization parameters depending on the approximation used in space. For the pressure they prove optimal stability and convergence only in the case of piecewise affine approximation under the standard condition on the time-step. These results have been reported in [39] .

Motivated by the results reported in [19] ,
E. Burman, A. Ern (ENPC) and M.A. Fernández have investigated the fully explicit treatment of the stabilization and the advection terms using
Runge-Kutta based methods. They have analyzed (second- and third-order) explicit Runge-Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first-order linear differential operator in space of Friedrichs-type. They establish L^{2} -norm error estimates with (quasi-)optimal convergence rates for smooth solutions in space and time. These results hold under the usual CFL condition for third-order Runge-Kutta schemes and any polynomial degree in space and for second-order Runge-Kutta schemes and first-order polynomials in space. For second-order Runge-Kutta schemes and higher polynomial degrees in space, a tightened 4/3 -CFL condition is required. These theoretical results and some numerical experiments have been reported in [38] .

#### Kinetics models

Participant : Laurent Boudin.

Kinetics models are typically used in our team to model aerosol in the respiratory tracts. In recent works, L. Boudin has also used them in the context of Sociophysics.

Sociophysics is a research field first introduced in the pioneering paper [65] in the early eighties. The basic idea is the fact that methods and concepts from physics can be used to describe political or social behaviors. A lot of works has been recently devoted to sociophysics in the sociological, physical and mathematical communities. L. Boudin and F. Salvarani (Univ. Pavia, Italy) started from the assessment that the tools of statistical physics fitted the sociophysical framework. More precisely, individuals are considered as particles whose *collective* behavior can be described through kinetic models.

In [16] , [17] , they studied a kinetic model of opinion formation where “binary interactions” between individuals (collisions) and “self-thinking” (diffusion) are taken into account. They obtained mathematical results (a priori estimates, existence, long-time behaviour) and performed numerical simulations on relevant situations, recovering results from other non kinetic models and obtaining original behaviors in new situations. In [36] , with R. Monaco (Politec. Torino, Italy), they proposed a multidimensional opinion formation model with the presence of media. They supervised a Master 2 internship on this kind of model with the addition of contradictory individuals in the population, and the related paper is currently being written. Eventually, they wrote a review chapter on kinetic models for opinion formation, in a book from the “Modeling and Simulation in Science, Engineering and Technology” Series, Birkhauser.