Section: Research Program

Computational fluid mechanics: high order discretization on unstructured meshes and efficient methods of solution

All the methods considered in the project are mesh-based methods: the computational domain is divided into cells, that have an elementary shape: triangles and quadrangles in two dimensions, and tetrahedra, hexahedra, pyramids, and prisms in three dimensions. If the cells are only regular hexahedra, the mesh is said to be structured. Otherwise, it is said to be unstructured. If the mesh is composed of more than one sort of elementary shape, the mesh is said to be hybrid.

The AeroSol library developed in the team is based on discontinuous Galerkin methods. These methods were introduced by Reed and Hill [38] and first studied by Lesaint and Raviart [36] . The extension to the Euler system with explicit time integration was mainly led by Shu, Cockburn and their collaborators. The steps of time integration and slope limiting were similar to high order ENO schemes, whereas specific constraints given by the finite element nature of the scheme were progressively solved, for scalar conservation laws [29] , [28] , one dimensional systems [27] , multidimensional scalar conservation laws [26] , and multidimensional systems [30] . For the same system, we can also cite the work of [32] , [35] , which is slightly different: the stabilization is made by adding a nonlinear term, and the time integration is implicit.

Contrary to continuous Galerkin methods, the discretization of diffusive operators is not straightforward. This is due to the discontinuous approximation space, which does not fit well with the space function in which the diffusive system is well posed. A first stabilization was proposed by Arnold [20] . The first application of discontinuous Galerkin methods to Navier-Stokes equations was proposed in [24] by mean of a mixed formulation. Actually, this first attempt led to a non compact computation stencil, and was later proved to be not stable. A compactness improvement was made in [25] , which was later analyzed, and proved to be stable in a more unified framework [21] . The combination with the $k-\omega$ RANS model was made in [23] . As far as Navier Stokes equations are concerned, we can also cite the work of [34] , in which the stabilization is closer to the one of [21] , the work of [37] on local time stepping, or the first use of discontinuous Galerkin methods for direct numerical simulation of a turbulent channel flow done in [31] . Discontinuous Galerkin methods are so popular because

• They can be developed for any order of approximation.

• The computational stencil of one given cell is limited to the cells with which it has a common face. This stencil does not depend on the order of approximation. This is a pro, compared for example with high order finite volumes, which require as more and more neighbors as the order increases.

• They can be developed for any kind of mesh, structured, unstructured, but also for aggregated grids [22] . This is a pro compared not only with finite differences schemes, which can be developed only on structured meshes, but also compared with continuous finite elements methods, for which the definition of the approximation basis is not clear on aggregated elements.

• $p$-adaptivity is easier than with continuous finite elements, because neighboring elements having a different order are only weakly coupled.

• Upwinding is as natural as for finite volumes methods, which is a benefit for hyperbolic problems.

• As the formulation is weak, boundary conditions are naturally weakly formulated. This is a benefit compared with strong formulations, for example point centered formulation when a point is at the intersection of two kinds of boundary conditions.

For concluding this section, there already exist numerical schemes based on the discontinuous Galerkin method which proved to be efficient for computing compressible viscous flows. Nevertheless, there remain many things to be improved, which include: efficient shock capturing methods for supersonic flows, high order discretization of curved boundaries, low Mach number behavior of these schemes and combination with second-moment RANS models. Another drawback of the discontinuous Galerkin methods is that they can be computationally costly, due to the accurate representation of the solution calling for a particular care of implementation for being efficient. We believe that this cost can be balanced by the strong memory locality of the method, which is an asset for porting on emerging many-core architectures.