Many aspects of the large-scale instabilities that appear in a magnetic confined plasma can well be described in the magneto-hydrodynamic (MHD) framework (with additional physics extensions). Solving these equations globally in the complex geometry of a divertor tokamak or a stellarator is a highly demanding task due to the strong temporal and spatial multi-scale nature of the problem and highly anisotropic behaviour arising from strong magnetic fields.
The research topics addressed are connected to the mathematical modeling and simulation of MHD equations, including:
The MHD group is mainly working in High Order Continuous and Discontinuous Galerkin Finite Elements methods
- Isoparametric and Isogeometric Analysis (IGA): IGA allows the use of Computer Aided Design (CAD) shape functions to approach the unknowns of a (system of) partial differential equation(s). The geometry is then given by a geometric transformation (mapping) that maps a given logical domain onto the physical one.
- Compatible Finite Elements Methods: Developing numerical methods and codes that allow the construction of different functional spaces, including H(div) and H(curl) spaces.
- High Order Discontinuous Galerkin Method: DG methods represent the solution by element-local polynomials and discontinuities at element interfaces are resolved via unique numerical fluxes. The scheme is conservative and high order. Due to its locality, DG is highly scalable and therefore well suited for large-scale computations. The MHD group actively develops DG methods for MHD simulations.
- Mesh Generation: The strong anisotropy in MHD simulations requires that meshes must follow the topology of the magnetic field. In order to achieve realistic simulations in both tokamaks and stellarators, meshes have to be generated with the knowledge of an MHD equilibrium or even by coupling to an equilibrium solver. In addition, the tools developed in the MHD group aim to generate meshes that are taylored to match discrete properties of each numerical scheme.
Fast solvers and robust preconditioners
Due to their multi-scale nature, MHD equations are in general solved implicitly in time.The critical and most expensive point is the solution of the linear systems. In fact, the classical preconditioners fail to converge in realistic simulations while the use of direct solvers is limited by the memory consumption. In order to overcome this limitation, the development of physics based preconditioners and a new class of preconditioners based on the Generalized Locally Toeplitz thoery (GLT) and being attemped. The GLT allows us to understand the spectral pathology of the obtained linear systems. Moreover, using the GLT, fast preconditioners can be derived, especially for B-Splines discretizations.
Hybrid Particles In Cell and Semi-Lagrangian Methods
Together with the Kinetic group, the MHD group is developing hybrid algorithms for Particles In Cell and Semi-Lagrangian methods. Here, "hybrid" refers to the representation of particle positions in logical space while keeping particle velocities in physical space. This allows to treat complex geometries with a reasonable cost.
An important mission of the MHD group is to provide parallel, robust and efficient softwares for the numerical simulation of the MHD equations:
- FLEXI / HOPR: The FLEXI solver implements the high order Discontinuous Galerkin Spectral Element Method on abritrarily shaped 3D hexahedral meshes. The FLEXI solver is fully MPI parallelized and scales linearly up to thousands of cores. Several equation systems are implemented, from anisotropic diffusion to ideal and resistive MHD. The meshes for FLEXI are generated with HOPR(high order preprocessor). HOPR has an interface to VMEC data to realize meshes of both tokamak and stellarator equilibria. HOPR & FLEXI are developed in cooperation with theInstitute of Aerodynamics at the University of Stuttgart and the Mathematical Institute at the University of Cologne.
- Jorek-Django: is a Finite Elements framework. It is developed together with INRIA Nancy Grand-Est and INRIA Sophia-Antipolis. Jorek-Django aims to offer different kind of Finite Elements discretizations (B-Splines/NURBS, Hermite-Bézier, Powell/Sabin, Fourier). Jorek-Django allows the user to develop and construct physics based preconditioners. In the case of B-Splines, it also offers GLT based preconditioners.