Séminaires A3 à venir

Les théorèmes de nonexistence et de classification de type Liouville, pour les problèmes elliptiques non linéaires dans l'espace ou le demi-espace, ont une longue histoire, avec des contributions classiques de Bernstein, de Giorgi, Gidas-Spruck, Berestycki-Caffarelli-Nirenberg, ... D'autre part le système de Lane-Emden (LE) -\Delta u=v^p, -\Delta v=u^q avec p, q>1, est un exemple modèle de système elliptique hamiltonien et a été intensivement étudié. Nous présenterons un résultat récent de type Liouville qui établit la nonexistence de solutions positives de (LE) dans le demi-espace avec conditions nulles au bord. La nouveauté du résultat est l'absence de restrictions sur p, q et sur la croissance de la solution lorsque tend vers l’infini (en collaboration avec Yimei Li)

Excited state molecular dynamics (MD) simulations are a powerful computational tool for the study of photoinduced phenomena. A popular method to get the excited state energy and its gradients is linear response time dependent density functional theory (TD-DFT). In this method, the excitation energies and the corresponding perturbations to the electronic density are obtained as eigenvalues and eigenvectors of the molecular response function. The gradients are then used to propagate the equations of motion of the molecule. At each step of an MD simulation, regardless of the method, the new molecular geometry is relatively close to the previous ones. This observation suggests that some extrapolation strategy can be applied, such that the results of the previous calculations (which are available for free) can be used to predict the result of the upcoming calculation. The prediction can then be used as a guess for the iterative solver to lower the number of iterations and thus the cost. In the case of ground state MD simulations, several strategies extrapolation strategies have been proposed. Among these, the most successful are the extended Lagrangian from Niklasson[1], the GExt scheme from my coworkers[2,3] and Car-Parrinello MD strategies[4]. In the case of linear response TD-DFT MD, only a strategy based on the extended Lagrangian has been proposed so far[5]. In this talk, I first introduce the linear response TD-DFT method for excited states, highlighting its connection with differential geometry. Then, I present an extension of the GExt scheme presented in [3] to linear response TD-DFT, in which the manifold structure to which the solutions belong is used to make the extrapolation more accurate. Lastly, I show how the new extrapolation strategy can be applied to excited state MD simulations resulting in a significant acceleration. [1] Niklasson, J. Chem. Phys. 152 (2020). [2] Polack et al., J. Chem. Theory Comput. 17 (2021). [3] Pes et al., J. Phys. Chem. Lett. 14 (2023). [4] Kühne, WIREs Comput. Mol. Sci 4 (2014). [5] Niklasson et al., J. Chem. Theory Comput. 14 (2018).

In the first part of this talk, we present a high-order pressure-based solver for the 3D compressible Navier–Stokes equations operating uniformly at all Mach numbers. The method uses a cell-centered split of fast and slow fluxes, treated implicitly and explicitly, and introduces semi-implicit discretizations of kinetic energy and enthalpy that avoid iterative solvers. The resulting scheme leads to an elliptic pressure equation suitable for ideal and general EOS, solved with a nested Newton method when needed. High-order accuracy is achieved through IMEX time stepping and a dimension-by-dimension CWENO reconstruction for explicit convective and viscous terms. The implicit part uses central schemes without added dissipation, yielding a CFL condition based only on fluid velocity, and ensuring asymptotic preservation in the low-Mach limit. The second part presents an implicit DG-based discretization of the viscous terms, producing a discrete Laplacian built from high-order corner gradients in 3D and yielding a symmetric, positive-definite system. The final part introduces a Structure-Preserving DG (SPDG) operator that enforces the discrete div–curl identity to machine precision on staggered Cartesian grids. We illustrate its performance through simulations of incompressible Navier–Stokes equations in vortex–stream formulation and validate the robustness of the overall approach on a broad set of inviscid and viscous benchmark problems across all Mach regimes.

TBA