## Section: New Results

### Numerical methods for fluid flows

#### Kinetic entropy for the layer-averaged hydrostatic Navier-Stokes equations

Participants : Emmanuel Audusse, Marie-Odile Bristeau, Jacques Sainte-Marie.

In [26], the authors are interested in the numerical approximation of the hydrostatic free surface incompressible Navier-Stokes equations. By using a layer-averaged version of the equations, previous results obtained for shallow water system are extended. A vertically implicit / horizontally explicit finite volume kinetic scheme is designed that ensures the positivity of the approximated water depth, the well-balancing and a fully discrete energy inequality.

#### Numerical approximation of the 3d hydrostatic Navier-Stokes system with free surface

Participants : Marie-Odile Bristeau, Anne Mangeney, Jacques Sainte-Marie, Fabien Souillé.

*In collaboration with S. Allgeyer, M. Vallée, R. Hamouda, D. Froger*.

A stable and robust strategy is proposed to approximate incompressible hydrostatic Euler and Navier-Stokes systems with free surface. The idea is to use a Galerkin type approximation of the velocity field with piecewise constant basis functions in order to obtain an accurate description of the vertical profile of the horizontal velocity. We show that the model admits a kinetic interpretation, and we use this result to formulate a robust finite volume scheme for its numerical approximation.

#### Well balanced schemes for rotation dominated flows

Participants : Emmanuel Audusse, Do Minh Hieu, Yohan Penel.

*In collaboration with P. Omnes.*

In [27], we study the property of colocated Godunov type finite volume schemes applied to the linear wave equation with Coriolis source term. The purpose is to explain the bad behaviour of the classical scheme and to modify it in order to avoid accuracy issues around the geostophic equilibrium. We use tools from two communities: well-balanced schemes for the shallow water equation with topography and asymptotic preserving schemes for the low Mach model. CFL conditions that ensure the stability of fully discrete schemes are established. The extension to the nonlinear case is under study.

#### A two-dimensional method for a dispersive shallow water model

Participants : Nora Aïssiouene, Marie-Odile Bristeau, Anne Mangeney, Jacques Sainte-Marie.

*In collaboration with C. Pares*.

In [29], [6], we propose a numerical method for a two-dimensional dispersive shallow water system with topography [3]. A first approach in one dimension, based on a prediction-correction method initially introduced by Chorin-Temam has been presented in [33]. The prediction part leads to solving a shallow water system for which we use finite volume methods while the correction part leads to solving a mixed problem in velocity and pressure. From the variational formulation of the mixed problem proposed in [35], the idea is to apply a finite element method with compatible spaces to the two-dimensional problem on unstructured grids.

#### Entropy-satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

Participant : Martin Parisot.

Article [32] is devoted to the numerical resolution in multidimensional framework of a hierarchy of reduced models of the free surface Euler equations. An entropy-satisfying scheme is proposed for the monolayer dispersive models [40] and [3]. To illustrate the accuracy and the robustness of the strategy, several numerical experiments are performed. In particular, the strategy is able to deal with dry areas without particular treatment. A work in progress focuses on the adaptation of the entropy-satisfying scheme to the layerwise models proposed in [30].

#### A lateral coupling between river channel and flood plain with implicit resolution of shallow water equations

Participant : Martin Parisot.

*In collaboration with S. Barthélémy, N. Goutal, M.H. Le, S. Ricci*.

Multi-dimensional coupling in river hydrodynamics offers a convenient solution to properly model complex flow while limiting the computational cost and taking the advantage of most pre-existing models. The project aims to adapt the lateral interface coupling proposed in [39] to the implicit version and assess it with real data from the Garonne River.

#### The discontinuous Galerkin gradient discretisation

Participant : Cindy Guichard.

*In collaboration with R. Eymard*.

The Symmetric Interior Penalty Galerkin (SIPG) method, based on Discontinuous Galerkin approximations, is shown to be included in the Gradient Discretisation Method (GDM) framework. Therefore, it can take benefit from the general properties of the GDM, since we prove that it meets the main mathematical gradient discretisation properties on any kind of polytopal mesh. We illustrate this inheritance property on the case of the $p$−Laplace problem [13].

#### Gradient-based optimization of a rotating algal biofilm process

Participants : Pierre-Olivier Lamare, Jacques Sainte-Marie.

*In collaboration with N. Aguillon, O. Bernard*.

Here we focus on the optimal control of an innovative process where the microalgae are fixed on a support. They are thus successively exposed to light and dark conditions. The resulting growth can be represented by a dynamical system describing the denaturation of key proteins due to an excess of light. A PDE model of the Rotating Algal Biofilm is then proposed, representing local microalgal growth submitted to the time varying light. An adjoint-based gradient method is proposed to identify the optimal (constant) process folding and the (time varying) velocity of the biofilm.

#### Method of reflections

Participant : Julien Salomon.

*In collaboration with G. Legendre, P. Laurent, G. Ciaramella, M. Gander, L. Halpern*.

In [17], the authors carefully trace the historical development of the methods of reflections, give several precise mathematical formulations and an equivalence result with the alternating Schwarz method for two particles.

In [31], a general abstract formulation is proposed in a given Hilbert setting and the procedure is interpreted in terms of subspace corrections. The unconditional convergence of the sequential form is proven and a modification of the parallel one is proposed to make it unconditionally converging.