Development of a gyrokinetic hyperbolic solver based on discontinuous Galerkin method in tokamak geometry

Abstract

A hyperbolic solver is developed for the gyrokinetic equation in tokamak geometry. Aiming a whole device modeling of fusion plasma surrounded by first walls of tokamak device, an unstructured spatial mesh is introduced. The discontinuous Galerkin (DG) method is used to discretize the gyrokinetic equation on the mesh and test various numerical elements for the discretization. Based on the conservations of physical quantities such as mass, kinetic energy, and toroidal canonical angular momentum in an axisymmetric configuration of toroidal plasma, we investigate the effects of basis functions for the DG method on the numerical solutions. With proper choices of the basis functions and spatial grid resolutions, the conservations of the key physical quantities are shown to be satisfied nearly to machine accuracies in the simplified circular magnetic geometry. Even in realistic tokamak geometry with machine wall boundaries, it is shown that a good conservation property can be demonstrated if the flux across boundaries of a test domain is carefully accounted for. Also, the invariance of the canonical Maxwellian distribution function in time is well satisfied with the developed solver. The effect of weighting functions for the basis is investigated too. Overall, the Maxwellian weighted basis shows a similar conservation property with the polynomial basis. On the other hand, the Maxwellian weighted basis shows better performance in resolving small scale structures in velocity space, which can be utilized to set up an efficient basis set to simulate fine structures with less computational costs. The parallelization of the newly developed solver is also reported. Employing MPI for the parallelization, the solver shows good performances up to a few thousand CPU cores.