Kinetic and Gyrokinetic Models
Kinetic theory gives a probabilistic description of the plasma in terms of a distribution function in phase-space. This description is more comprehensive than the magneto-hydrodynamic description but at the same time more computationally demanding. A major challenge for the numerical solution of the kinetic model is the rather high dimensionality of four to six dimensions depending on the geometry and the level of approximation. In addition the requirements on the resolution is high since small filaments and turbulent structures can occur in phase-space. The kinetics group of the NMPP section works on the development and analysis of robust and efficient algorithms for numerical simulations of the kinetic equations, in its most general six-dimensional form as well as the gyrokinetics equations. Mostly particle-in-cell and semi-Lagrangian methods are developed, but Eulerian methods based on spectral discontinuous Galerkin are also under scrutiny. Moreover, the group works with the analysis of various approximations in the model and on verification.
In fusion plasmas the strong magnetic field allows the fast gyro-motion to be systematically removed from the description of the dynamics, resulting in a considerable model simplification (from 6 towards 4 dimensional particle phase space). One major goal is to develop a modern gyrokinetic code for tokamak as well as stellarator geometries that is capable of simulating the physics close to the X-point and the magnetic axis. Efficient methods needs to be devised that are also adapted to the physics. For instance interpolation methods that are aligned with the magnetic field lines are designed to account for the fact that the particles - and hence also structures in the distribution functions - mostly follow the field lines. In order to be able to compute on realistic geometries, mesh generation is of major importance. Mesh definition with tools from isogeometric analysis and the combination of several patches are currently under development.
Hybrid fluid-kinetic models
Together with the fluid group, hybrid methods are designed that combine kinetic and fluid models, for instance in order to combine fast and slow particles or to accelerate kinetic codes. Properly understanding the transition from the kinetic to the fluid model is another research goal.
Verification is an important effort in the gyrokinetics community. Advanced mathematical tools such as variational formulation of dynamics are used in order to create a hierarchy of existing models implemented into different European gyrokinetics codes and to classify them according to their complexity and physical properties.
Fully kinetic description
In addition to the projects on efficient numerical methods for the state-of-the-art gyrokinetic models, solvers for a fully kinetic description in six-dimensional phase-space are also developed. In order to be able to use grid-based methods in six-dimensional phase space, semi-Lagrange methods are designed that incorporate compression via sparse grids or hierarchical tensors. The classical approach for a six-dimensional solution are particle-based methods. Compared to grid-based methods, particle algorithms scale better in higher dimensions, but suffer from rather strong noise. Methods from stochastic analysis are used to quantify and reduce the noise. Besides classical particle-in-cell algorithms, particle-in-Fourier methods are of interest since they are specially suited for phenomena that are characterized by a few important wave lengths.
Together with the Tonus (tokamaks and numerical simulations) INRIA team and other French partners the Fortran library SeLaLib is developed, a modular software framework for (gyro)kinetic simulations. In addition, the group also cooperates with developers of state-of-the-art fusion simulation codes like the electromagnetic ORB5, GENE and Gysela on the validation of the underlying physical models or the improvement of their numerical schemes.