## Section: New Results

### Numerical methods for fluid flows

Participants : Jacques Sainte-Marie, Virgile Dubos, Cindy Guichard, Martin Parisot, Marie-Odile Bristeau, Fabien Souillé, Edwige Godlewski, Yohan Penel.

#### Advancing dynamical cores of oceanic models across all scales

Oceanic numerical models are used to understand and predict a wide range of processes from global paleoclimate scales to short-term prediction in estuaries and shallow coastal areas. One of the overarching challenges, and the main topic of the COMMODORE workshop, is the appropriate design of the dynamical cores given the wide variety of scales of interest and their interactions with atmosphere, sea-ice, biogeochemistry, and even societal processes. The construction of a dynamical core is a very long effort which takes years and decades of research and development and which requires a collaborative mixture of scientific disciplines. In [14], we present a significant number of fundamental choices, such as which equations to solve, which horizontal and vertical grid arrangement is adequate, which discrete algorithms allows jointly computational efficiency and sufficient accuracy, etc.

#### A Well-balanced Finite Volume Scheme for Shallow Water Equations with Porosity

Our work [20] aims to study the ability of a single porosity-based shallow water model to modelize the impact of vegetation in open-channel flows. More attention on flux and source terms discretizations are required in order to archive the well-balancing and shock capturing. We present a new Godunov-type finite volume scheme based on a simple-wave approximation and compare it with some other methods in the literature. A first application with experimental data was performed.

#### The gradient discretisation method

The monograph [21] is dedicated to the presentation of the gradient discretisation method (GDM) and to some of its applications. It is intended for masters students, researchers and experts in the field of the numerical analysis of partial differential equations.The GDM is a framework which contains classical and recent discretisation schemes for diffusion problems of different kinds: linear or non-linear, steady-state or time-dependent.

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

This work [29] is devoted to the numerical resolution in multidimensional framework of a hierarchy of reduced models of the free surface Euler equations. In a first part, entropy-satisfying scheme is proposed fo the monolayer dispersive model [Green, Naghdi '76] and [Bristeau, Mangeney, Sainte-Marie, Seguin '15]. In a second part, the strategy is extended to the layerwise models proposed in [Fernandez-Nieto, Parisot, Penel, Sainte-Marie]. 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.

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

In this work [23], we propose a stable and robust strategy to approximate the 3d incompressible hydrostatic Euler and Navier-Stokes systems with free surface. Compared to shallow water approximation of the Navier-Stokes system, 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.

#### Congested shallow water model: floating object

In [27], we are interested in the floating body problem on a large space scale. We focus on objects floating freely in the water such as icebergs or wave energy converters. The formulation of the fluid-solid interaction using the congested shallow water model for the fluid and Newton's second law of motion for the solid is given and a strong coupling between the two systems is explained. The energy transfer between the solid and the water is focused on since it is of major interest for energy production. A numerical resolution based on the coupling of a finite volume scheme for the fluid and a Newmark scheme for the solid is presented. An entropy correction based on an adapted choice of discretization for the coupling terms is made in order to ensure a dissipation law at the discrete level. Simulations are presented to validate the method and to show the feasibility of more complex cases.

#### Numerical strategies for a dispersive layer-averaged model

A hierarchy of models has been derived in [12] to approximate the Euler equations by means of a layer-averaging procedure. This results in several dispersive models with one velocity field per layer. The structure of the equations induces issues of efficiency. The standard splitting between hydrostatic and non-hydrostatic components leads to a prohibitive computational costs. In a work in progress, we are investigating a new strategy to solve the projection step in a cheaper way. This is assessed by means of steady nontrivial solutions of the dispersive equations.

#### Methods of Reflections

The basic idea of the method of reflections appeared almost two hundred years ago; it is a method of successive approximations for the interaction of particles within a fluid, and it seems intuitively related to the Schwarz domain decomposition methods, the subdomains being the complements of the particle domains. We show in [25] that indeed there is a direct correspondence between the methods of reflections and Schwarz methods in the two particle/subdomain case. This allows us to give a new convergence analysis based on maximum principle techniques with precise convergence estimates that one could not obtain otherwise. We then show however also that in the case of more than two particles/subdomains, the methods of reflections and the Schwarz methods are really different methods, with different convergence properties. We finally also introduce for the first time coarse corrections for the methods of reflections to make them scalable in the case when the number of particles becomes large.