Emmanuel Franck |
Email:
emmanuel.franck@inria.fr
Institut de Recherche Mathématique Avancée
7 Rue René Descartes,
67000 Strasbourg, France.
Asymptotic preserving schemes for linear transport on unstructured meshes. We consider hyperbolic systems with stiff source terms used to approximate the linear transport equation (or Boltzmann linear equation) present in neutronic or radiative transfer. These systems are dependent of a relaxation parameter which generate a diffusion limit. In this work we propose to design finite volumes schemes on unstructured meshes with convergence estimates and stability conditions independent of the relaxation parameter (asymptotic preserving methods). see more
Asymptotic preserving and Well-Balanced schemes for non linear hyperbolic systems in fluid mechanics. In this work we study some asymptotic limit the the behavior of the numerical schemes in these regimes. The first point concerns the asymptotic preserving positives schemes (methods with the convergence estimates and stability conditions are independent of the relaxation parameter) and "well-balanced" schemes (methods which preserve the steady states associated to the PDE) for the Euler equations with friction and gravity. The second limit studied is the low mach limit. see more
Implicit finite element schemes for MHD and reduced MHD. The context of this work is the resolution of the MHD equations (Jorek code) and the simulation of the plasma instabilities for the Tokamak as ITER. Firstly we propose to study the theoretical and numerical stability in time for the reduced MHD models. The second part concers the implicit time schemes for the MHD: full solver + preconditioning or splitted solver. The last part is about the compatigle finite element coupled with the implicit scheme. see more
Implicit high-order relaxation scheme for hyperbolic/parabolic PDE. The relaxation models allows to approximate a nonlinear system by a larger linear system with a nonlinear local source. Combining high order splitting scheme, and implicit for the linear hyperbolic systems and the source terms we obtain an high order implici schemes with simple implicit part. In this project, we construct method, proposed high order exntension, consistency and boudnary anaysis, parallel implementation and apply this to many applications. see more
Reduced models using deep learning for plasma physics. The context of this works is the construction of reduced model for Vlasov equation. We consider two type of models: asymptotic models (fluid models in collisional limit, gyro-kinetic models in large magnetic field limii) and reduced order modeling. In this project, we study the possibility to construct these models or a part of these models using recent deep learning method and try to assure tge stability. see more
Hybrid numerical methods with deep learning. The objective of this project is to design new numerical methods which coupling numerical scheme or solvers with neural networks using supervised or reinforcement deep learning. The applications will be: stabilization of nymerical schemes for conservation laws, design more accurate schemes or solver for hyperbolic/parabolic PDE. see more