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Development of numerical methods and mathematical models for the ICF

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Development of the ICF Lagrangian code CHIC

Since 2003, we are developing a two-dimensional Lagrangian radiation hydrodynamic code for ICF and other applications named CHIC. It is based on a new cell-centred unstructured Godunov-type scheme that solves the gas dynamics equations written in the Lagrangian formalism. This original approach leads to first-order conservative schemes that satisfy a local semi-discrete entropy inequality. The high-order extension of this scheme has been derived using a one-step time integrator based on the generalized Riemann methodology. The high-order cylindrical extension of this scheme is also implemented. The robustness and the accuracy of this scheme is assessed by running several validation test problems.

To improve, the robustness of the Lagrangian scheme in the CHIC, an ALE strategy has been developed. This methodology allows a computation to be broken up into a Lagrangian phase, a mesh rezoning phase intended to correct for mesh deformation and a remapping phase were computed physical values are conservatively interpolated from the distorted Lagrangian grid to the smoother rezoned grid. This framework enables to perform the full simulation of spherical target with hydrodynamic perturbations characterized by high wave numbers. To take into account heat conduction by electrons and photons, a specific numerical scheme for anisotropic diffusion has been implemented. This original finite volume scheme is able to compute accurately heat conduction even on highly distorted grid.
The following physical packages are implemented in CHC: laser propagation using a 3D ray-tracing method, radiative heat transport using the multi-group diffusion approximation, transverse magnetic field using a generalized Ohm’s law, electron heat transport using a non-local multi-group model, which includes magnetic field effect, thermonuclear burn in the diffusion approximation and a material database, which provides tabulated spectral opacities and equations of states.
The theoretical work on the Lagrangian hydrodynamics relies on several national (IMB, INRIA at Bordeaux, IMT at Toulouse, CEA-DIF at Bruyères Le Châtel) and international (LANL at Los Alamos, Sandia at Albuquerque and CTU at Prague) collaborations.

Development of kinetic Fokker-Planck code

We developed a new collisional kinetic and electromagnetic, massively parallel code describing the relativistic electrons in 2D in space and in 3D in the phase space. It serves as a reference code for various problems of fusion plasmas related to the energy transport by fast electrons. Efficient numerical methods (accurate, robust and fast) that account for the physical symmetries of the model have been implemented. The numerical scheme captures the discontinuities in the phase space. The CEA resources (CCRT-Platinum) are intensively used for the code validation in a wide range of parameters relevant to ICF.

For modelling of the realistic cases in a long (hundred picoseconds) time scale we are conducting a project "Kinetic Transport for Shock Ignition" on the GENCI computing center in collaboration with teams of the
Oxford University and Imperial College. It concerns the detailed simulations in 2D with two Fokker-Planck codes of the energy transport from the zone of laser energy absorption by fast electrons.

Development of a new particle-in-cell (PIC) code

Since 2008, with the arrival of a new Assistant Professor, E. d’Humières, we became users and active developers of a relativistic kinetic electromagnetic code in three dimensions named PICLS [Y. Sentoku and A. Kemp, J. Comp. Phys. 2007] dedicated to modelling of laser-plasma interaction at high laser intensity and short pulse durations. This is a 3D relativistic PIC code allowing to use different particles/fields interpolation orders. By applying a 3rd or 4th order interpolation, it is possible to extend the simulation grid size to the plasma skin length, to reduce the computational cost drastically and to perform extensive simulations of fast electron transport in dense plasmas.

An advantage of the PICLS over conventional PIC codes is in modeling plasma density gradients with variable weighted particles. Keeping the charge-to-mass ratio of the macroparticles constant, we can vary the density of real charges that they represent. This option provides a strong reduction os comuting resources in the problems where the density varies from the underdense region, where the laser pulse is absorbed, to the more than a few hundred times overdense target.
We have developed a fully relativistic binary collision model that assures the energy conservation in individual collisions and momentum conservation in average, which is a great advantage for high density plasma simulations, where the numerical heating or energy violation must be very small to describe accurately the laser coupling to plasmas. We showed in test simulations that the code reproduces the analytical exchange rate, confirming that the microscopic collision Monte-Carlo method reproduces the macroscopic quantities such as the average density and energy. We also confirmed that the fast electron stopping in dense plasmas in the range of energies from 10 keV to 1 GeV agrees with the NIST database, including electron-electron and electron-ion collisions.

Development of a mesoscopic code for laser-plasma interaction

The interpretation of experimental data and the necessity to perform predictive modeling requires a continuous effort to modify and upgrade existing codes used for microscopic and macroscopic laser-plasma interaction. The two-dimensional PIC-code used for the SBS amplification scenario was modified in collaboration with the CPHT in order to allow for counter propagating laser pulses. In the context of macroscopic laser-plasma interaction a new nonlinear hydrodynamic code was developed that substituted the existing Lagrangian module. This new hydrodynamic code is coupled with existing modules for the non-local transport and the code PARAX, describing the paraxial propagation of the laser beam. This new code is applied for interpretation of an experiment on electrostatic fields generated due to the propagation of a laser beam in large-scale plasma.
In collaboration with CEA/CESTA we developed a new self-consistent numerical method for treatment of the ionization processes in dielectrics in the femtosecond regime. It is based on the solution of the full 3D set of Maxwell’s equations completed by multiphoton or tunnel and collisional ionization model.

Mathematical models for the energy transport

In collaboration with the astrophysical team of CEA/Saclay we have developed a new method of computation the radiative transport. It is based on the maximum entropy closure approach, which consists in averaging the radiative intensity on angles. The closure of moment equations is obtained by using an underlying distribution function that assures the positiveness of the equilibrium function. The resulting hyperbolic multi-group system of partial differential equations is well-posed and recovers the known transport regimes such as the free-streaming and the black-body equilibrium.

The similar approach is applied for the electron transport. We derived a fast mesoscopic kinetic model that describes with a good precision and robustness the essential kinetic effects. This mesoscopic model is hyperbolic and assures the positivity of the density and a local dissipation of the entropy. This work is carried out in collaboration with the Institute Bergonié of Bordeaux, the Technical University of Kaiserslautern and the laboratory CENBG of Bordeaux.