Geometric multigrid for the gyrokinetic Poisson equation from fusion plasma applications

Kurzfassung

In order to face climate change and to preserve our ecosystem, we have to reduce the overall emission of carbon dioxide into the atmosphere. A promising addition to renewable energies is nuclear fusion. Delivering an almost infinite amount of clean and safe energy and with almost inexhaustible resources on earth, plasma fusion would solve all the world's climate and energy problems. However, being extremely complex, the reaction cannot be maintained for sufficient long time, yet, as it is extremely unstable. As the construction and operation of fusion reactors, e.g. tokamaks, is exceptionally expensive, numerical simulations are required in order to increase our knowledge about the fusion process. One existing code for plasma simulations in a tokamak is called GyselaX, in which a subroblem consists in solving a two dimensional Poisson equation on many cross-sections of the reactor geometry. The EoCoE (Energy Oriented Center of Excellence: toward exascale for energy) project, funded by the European Commission, aims for the improvement of the current solver for this equation in order to reduce the simulation times. In [1] and [2], a geometric multigrid approach using finite differences for the discretization and a combined line smoothing procedure has been developed. Additionally, an implicit extrapolation technique is used to increase the approximation order of the solution. In this master's thesis, this GmgPolar solver is detailed and implemented in C++. Moreover, several improvements have been applied to the solver and some parts of the code have been parallelised. As the full optimization and parallelisation exceeds the scope of this thesis, future work will be required, before comparing the solver with two other possible approaches and integrating it into GyselaX to reduce the simulation time. [1] Kühn, M. J.; Kruse, C.; Rüde, U. Energy-Minimizing, Symmetric Discretizations for Anisotropic Meshes and Energy Functional Extrapolation, SIAM J. Sci. Comput.Vol. 43(4), pp. A2448-A2473 (2021). [2] Kühn, M. J.; Kruse, C.; Rüde, U. Implicitly extrapolated geometric multigrid on disk-like domains for the gyrokinetic Poisson equation from fusion plasma applications, Preprint: https://hal.archives-ouvertes.fr/hal-03003307/, Submit-ted to Journal of Scientific Computing, 2021.