## Section: Research Program

### Optimizing the computational efficiency

#### High order nonlinear numerical methods

The numerical experiments carried out in [7] show that in case of very strong anisotropy, the convergence of the proposed NNM becomes too slow (less than first order). Indeed, the method appears to strongly overestimate the dissipation. In order to make the method more competitive, it is necessary to estimate the dissipation in a more accurate way. Preliminary numerical results show that second order accuracy in space can be achieved in this way. One also aims to obtain (at least) second order accuracy in time without jeopardizing the stability. For many problems, this can be done by using so-called two-step backward differentiation formulas (BDF2) [78].

Concerning the inhomogeneous fluid models, we aim to investigate new methods for the mass equation resolution.
Indeed, we aim at increasing the accuracy while maintaining some positivity-like properties and the efficiency
for a wide range of physical parameters. To this end, we will consider *residual distribution* (RD) schemes,
that appear as an alternative to finite volume methods. RD schemes enjoy very compact stencils. Therefore,
their extension from 2D to 3D yield reasonable difficulties. These methods appeared twenty years ago, but
recent extensions to unsteady problems [91], [84], with
high-order accuracy [54], [53], or for parabolic problems [51], [52]
make them very competitive. Relying on these breakthroughs, we aim at designing new RD schemes
for fluid mixture models with
high-order accuracy while preserving the positivity of the solutions.

*A posteriori* error control

The question of the *a posteriori* error estimators will also have to be addressed in this optimization context. Since the pioneering papers of Babuska and Rheinboldt more than thirty years ago [59], *a posteriori* error estimators have been widely studied.
We will take advantage of the huge corresponding bibliography database in order to optimize our numerical results.

For example, we would like to generalize the results we derived for the harmonic magnetodynamic case
(e.g. [10] and [70]) to the temporal magnetodynamic one, for which space/time
*a posteriori* error estimators have to be developed. A space/time refinement algorithm should consequently
be proposed and tested on academic as well as industrial benchmarks.

We also want to develop *a posteriori* estimators for the variable density Navier-Stokes model or some of its
variants. To do so, several difficulties have to be tackled: the problem is nonlinear, unsteady,
and the numerical method [5], [6] we developed combines features from finite elements and finite volumes.
Fortunately, we do not start from scratch.
Some recent references are devoted to the unsteady Navier-Stokes model in the finite element context [63], [96].
In the finite volume context, recent references deal with unsteady convection-diffusion equations [95], [57], [73] and [67].
We want to adapt some of these results to the variable density Navier-Stokes system, and to be able to design an efficient space-time remeshing algorithm.

#### Efficient computation of pairwise interactions in large systems of particles

Many systems are modeled as a large number of punctual individuals ($N$) which interact pairwise which means $N(N-1)/2$ interactions. Such systems are ubiquitous, they are found in chemistry (Van der Waals interaction between atoms), in astrophysics (gravitational interactions between stars, galaxies or galaxy clusters), in biology (flocking behavior of birds, swarming of fishes) or in the description of crowd motions. Building on the special structure of convolution type of the interactions, the team develops computation methods based on the Non Uniform Fast Fourier Transform [82]. This reduces the $O\left({N}^{2}\right)$ naïve computational cost of the interactions to $O(NlogN)$, allowing numerical simulations involving millions of individuals.