## Section: New Results

### Numerical methods

#### Entropy viscosity for conservation laws

Participants : Jean-Luc Guermond [ Texas A & M University ] , Richard Pasquetti.

Together with J.L. Guermond, Texas A & M University at College Station, we are developing a technique to compute solutions of non-linear hyperbolic problems [23] , [22] , [21] ,[16] The main idea is here to introduce a non-linear viscous term which is set up from the residual of the entropy equation associated to the considered PDE. This entropy viscosity method has been applied using various types of approximations, from finite element to Fourier via spectral elements. We checked that the approach preserves the approximation properties of the underlying numerical method. Tests have been carried out for challenging 2D scalar conservation laws as well as for the (compressible) Euler equations with very satisfactory results. Just like the SVV (see section 6.3.1 ) technique, this entropy viscosity method may be of interest for the LES of turbulent flows.

#### Spectral element approximations

Participant : Richard Pasquetti.

Complementary to the thesis of A. Bonnement that develops an approach based on finite volume and low-order finite element method, a high order spectral element/Fourier code is currently developed to solve a two-fluid modeling of plasma flow and turbulence in tokamaks, with application to ITER. The code is already operational for the ion and electron temperature equations, which are characterized by a strong anisotropy in the diffusivity.

We are also working on the development of spectral element approximations on unstructured meshes [17] , in particular based on the use of simplicial elements. Solvers have been developed for elliptic problems and are currently developed for the incompressible Navier-Stokes equations, in the frame of the thesis of Laura Lazar, using a projection technique to take care of the divergence free constraint. The final algebraic systems being ill-conditioned, we especially focus on the resolution techniques. They may be based on a Schur complement method, in order to solve only for the unknowns located on the "skeleton" of the mesh, or on a p-multigrid approach, thus providing at the lowest level a finite element coarse solver.

#### Low Mach number flows

Participant : Hervé Guillard.

Following a series of works on the behavior of upwind schemes in the low Mach number regime, we have studied finite volume cell centered upwind schemes in this context [13] . We have shown that the problem of the lack of convergence toward the solutions of the incompressible system disappears and that convergence toward the incompressible solution is recovered. This result is given for arbitrary unstructured meshes. In addition, we also show that this result is equally valid for unstructured 3D tetrahedral meshes.

#### Mesh adaptation Methods

Participants : Anca Belme [ Projet Tropics ] , Alain Dervieux, Frédéric Alauzet [ Projet Gamma, INRIA-Rocquencourt ] , Adrien Loseille [ George washinston University ] , Damien Guégan [ Lemma ] .

In cooperation with Gamma , Tropics , pumas , a new adjoint-based mesh adaptation criterion has been developed and applied to sonic boom mitigation, for the HISAC European project and to several other CFD problems. See the comments of Tropics activity reports. A special effort was applied to mesh-adaptive capture of discontinuities in CFD. Steady problems with compressible shocks have been considered, in order to ensure second-order convergence despite of the discontinuities, see [19] . Unsteady discontinuities have been also considered, see the section 6.1.5 .

#### Flows with interfaces

Participants : Alain Dervieux, Hervé Guillard, Frédéric Alauzet [ Projet Gamma, INRIA-Rocquencourt ] , Olivier Allain [ Lemma ] , Damien Guégan [ Lemma ] , Thomas Bouchérès [ Lemma ] , Cécile Lesage [ Barcelona Computing Center ] .

The level set method is extended to new applications. Two fields of applications are considered, the interaction of sea surface with obstacles, and the motion of fuel in spacial tanks. A first topic particularly addressed in 2008-2009 is the improvement of mass conservation in the Level Set method. In [29] , we describe a new conservative formulation of Level Set, the Dual Level Set scheme. The main idea is to measure in a variational integral the defect of a predictor with respect to the advection of the discontinuous phase colour function. This is possible thanks to the continuous test functions used in the level set approximation scheme. A second important topic is the combination of a Level Set based Navier-Stokes numerical model with the fixed point dynamic mesh adaptive algorithm. As a result of the collaboration between Lemma , Gamma , and pumas , a paper presenting a method for this combination has been submitted.

#### Fluid-Structure interaction

Participants : Alain Dervieux, Charbel Farhat [ Stanford University ] , Bruno Koobus [ University of Montpellier 2 ] , Mariano Vàzquez [ Barcelona Supercomputing center ] .

The Geometric Conservation Law (GCL) expresses the exactness of an Arbitrary Lagrangian-Eulerian discretisation for uniform flows. We have demonstrated that this is a necessary condition for total energy conservation. This also extends the GCL to boundaries in a canonical manner. Total energy conservation is a key property for numerical models of any mechanical system in which the internal energy of a compressible fluid is converted into mechanical energy transmitted to a structure. A new finite-volume scheme satisfying this condition has been built from our previous scheme, developed and tested. The stabilisation effect of this improvement has been put in evidence for a standard flutter test case. The gain in accuracy has been evaluated for the motion of a piston in a closed vessel. A paper written in cooperation with Charbel Farhat (Stanford) and Mariano Vázquez (Barcelona Supercomputing Center) is submitted.

#### Parallel solvers for CFD algorithms

Participants : Hubert Alcin [ Tropics ] , Olivier Allain [ Lemma ] , Anca Belme [ Tropics ] , Marianna Braza [ IMF-Toulouse ] , Alain Dervieux, Bruno Koobus [ Université Montpellier 2 ] , Hilde Ouvrard [ University of Montpellier 2 ] , Stephen Wornom [ Lemma ] .

The parallel efficiency of our CFD algorithm has motivated many of our algorithmic investigations in the past. A careful evaluation of the performance of the Restrictive Additive Schwarz domain decomposition algorithm applied to high Reynolds RANS-LES calculations has been performed and presented in [31] .

pumas is associated to the ANR ECINADS project devoted to the design of new solution algorithms for unsteady compressible flows, adapted to scalable parallelism and to reverse (adjoint) Automatic Differentiation.