## Section: New Results

### Numerical analysis and simulation of heterogeneous systems

Participant : Xavier Antoine.

**Acoustics**

Artificial boundary conditions: while high-order absorbing boundary conditions (HABCs) are accurate for smooth fictitious boundaries, the precision of the solution drops in the presence of corners if no specific treatment is applied. In [29], the authors present and analyze two strategies to preserve the accuracy of Padé-type HABCs at corners: first by using compatibility relations (derived for right angle corners) and second by regularizing the boundary at the corner. Exhaustive numerical results for two- and three-dimensional problems are reported in the paper. They show that using the compatibility relations is optimal for domains with right angles. For the other cases, the error still remains acceptable, but depends on the choice of the corner treatment according to the angle.

Domain decomposition : in [49], Xavier Antoine and his co-authors develop the first application of the optimized Schwarz domain decomposition method to aeroacoustics. Highly accurate three-dimensional simulations for turbofans are conducted through a collaboration with Siemens (ongoing CIFRE Ph.D. Thesis of Philippe Marchner). In [26], the authors propose a new high precision IGA B-Spline approximation of the high frequency scattering Helmholtz problem, which minimizes the numerical pollution effects that affect standard Galerkin finite element approaches.

**Underwater acoustics**

New adiabatic pseudo-differential models as well as their numerical approximation are introduced in [53] for the simulation of the propagation of wave fields in underwater acoustics. In particular, the calculation of gallery modes is shown to be accurately obtained. This work is related to a new collaboration with P. Petrov from the V.I. Il'ichev Pacific Oceanological Institute, Vladivostok, Russia.

**Quantum theory**

With E. Lorin, Xavier Antoine proposes in [13] an optimization technique of the convergence rate of relaxation Schwarz domain decomposition methods for the Schrödinger equation. This analysis is based on the use of microlocal analysis tools. Convergence proofs are given in [11] for the real-time Schrödinger equation with optimized transmission conditions. We extend these results to the case of multiple subdomains in [13].

In [52], the authors analyze the convergence and stability in of a discretization scheme for the linear Schrödinger equation with artificial boundary conditions.

In [39], Xavier Antoine and his co-authors develop an implementation of the PML technique in the framework of Fourier pseudo-spectral approximation schemes for the fast rotating Gross-Pitaevskii equation. This is the first work related to the international Inria team BEC2HPC, associated with China (https://team.inria.fr/bec2hpc/).

In [12], Antoine and his co-authors develop new Fourier pseudo-spectral schemes including a PML for the dynamics of the Dirac equation. The implementation of the method leads to the possibility of simulating complex quantum situations. In [38], the authors extend the approximation to the curved static Dirac equation. The goal is to be able to better understand quantum phenomena related to the charge carriers in strained graphene, with potential long term applications for designing quantum computers. This is a collaboration with E. Lorin (Carleton University), F. Fillion-Gourdeau and S. Mac Lean from the Institute for Quantum Computing, University of Waterloo.

**Fractional PDE **

In [32], with J. Zhang and D. Li, Xavier Antoine is interested in the development and analysis of fast second-order schemes to simulate the nonlinear time fractional Schrödinger equation in unbounded domains.

The authors propose in [14] the construction of PML operators for a large class of space fractional PDEs in one- and two-dimensions. The specific case of the fractional laplacian is carefully considered.

Xavier Antoine and Emmanuel Lorin are interested in [40] in the problem of building fast and robust linear algebra algorithms based on the discretization of the Cauchy integral formula used to represent the power matrix. Applications related to stationary PDEs are presented, with possibly randomly perturbed potentials. Differential doubly preconditioned iterative schemes are investigated in details in [41] to evaluate the power, and more generally functions, of matrices.

Error estimates of a semi-implicit ALE scheme for the one-phase Stefan problem. J.F. Scheid, M. Bouguezzi (PhD student) and D. Hilhorst study the convergence with error estimates of an Arbitrary-Lagrangian-Eulerian (ALE) scheme for the classical one-phase Stefan problem. Despite Stefan problems as well as ALE techniques are well-known in the mathematical litterature for many decades, surprisingly there is no global result on convergence (with error estimates) for fully space-time discretized scheme based on ALE formulations. The main difficulty lies on the unbounded behavior of the exact (and approximate) free boundary. Stability results have already been obtained for a time-discretized scheme (and continuous in space) for the one-space dimension case.

Chaotic advection in a viscous fluid under an electromagnetic field. J.F. Scheid, J.P. Brancher and J. Fontchastagner study the chaotic behavior of trajectories of a dynamical system arising from a coupling system beetwen Stokes flow and an electromagnetic field. They consider an electrically conductive viscous fluid crossed by a uniform electric current. The fluid is subjected to a magnetic field induced by the presence of a set of magnets. The resulting electromagnetic force acts on the conductive fluid and generates a flow in the fluid. According to a specific arrangement of the magnets surrounding the fluid, vortices can be generated and the trajectories of the dynamical system associated to the stationary velocity field in the fluid may have chaotic behavior. The aim of this study is to numerically show the chaotic behavior of the flow for the proposed disposition of the magnets along the container of the fluid. The flow in the fluid is governed by the Stokes equations with the Laplace force induced by the electric current and the magnetic field. An article is in preparation.