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

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

A new gyrokinetic hyperbolic solver with discontinuous Galerkin method in tokamak geometry

We develop a hyperbolic solver for the gyrokinetic equation in tokamak geometry. The new solver is based on the discontinuous Galerkin approach on a finite element mesh composed of irregular spatial and regular velocity elements together with a strong-stability-preserving time discretization method. We investigate the effects of the basis function on the conservation properties of physical quantities such as mass, kinetic energy, and toroidal canonical angular momentum in an axisymmetric configuration of toroidal plasma. It is shown that if the proper basis function is chosen, the new solver has a good conservation property of the key physical quantities in the simplified circular magnetic geometry and realistic tokamak geometry. The invariance of the canonical Maxwellian distribution function in time is confirmed. We also investigate the effect of weighting functions for the polynomial basis. The weighted basis functions show a similar conservation property to the polynomial basis; the canonical Maxwellian weighted basis shows better invariance with the lower order polynomials. The performance tests of MPI parallelization are also carried out. The results indicate that the new solve solver performs well up to a few thousand CPU cores. [1] G. Jo, J.-M. Kwon, J. Seo, E. Yoon, Comput. Phys. Commun. 273 (2022) 10826

Nonlinear Fokker-Planck collision operator in Rosenbluth form for gyrokinetic simulations using discontinuous Galerkin method

A gyroaveraged nonlinear collision operator is formulated based on the Fokker-Planck operator in the Rosenbluth-MacDonald-Judd (RMJ) potential form and implemented for the gyrokinetic simulations with the discontinuous Galerkin scheme. The divergence structure of the original RMJ form is carefully preserved throughout the formulation to guarantee the density conservation while neglecting the finite Larmor radius effect. The B-spline finite element method is used to calculate the Rosenbluth potentials for the nonlinear collision operator. In addition to the nonlinear collision operator, linear and Dougherty collision models are also implemented to assess the benefits and drawbacks of each model. For the conservation of the parallel momentum and energy, we adopt a simple advection-diffusion model which numerically enforces the conservation of physical quantities. From bump-on-tail relaxation tests, the monotonically increasing entropy in time and conservation properties are demonstrated for the developed collision operator. Also, a few theoretical predictions for the neoclassical physics such as the neoclassical heat flux, poloidal flow and collisional damping of zonal flow are successfully reproduced by numerical simulations

Cost Assessment of a Tokamak Fusion Reactor with an Inventive Method for Optimum Build Determination

An inventive method was applied to determine the minimum major radius, R0, and the optimum build of a tokamak fusion reactor that simultaneously meets all physics, engineering, and neutronics constraints. With a simple cost model, tokamak systems analyses were carried out over ranges of system parameters to find an optimum build of a tokamak fusion reactor at minimum cost. The impact of a wide range of physics parameters and advanced engineering elements on costs were also addressed. When a central solenoid was used to ramp up a plasma current, design solutions with a cost of electricity (COE) between 109 and 140 mills/kWh, direct capital cost between 5000 and 6000 M/USD, and net electric power, Pe between 1000 and 1600 MW could be found with a minimum R0 between 6.0 and 7.0 m, and fusion power, Pfusion between 2000 and 2800 MW. When the plasma current was driven by a non-inductive external system, the system size and costs could be reduced further; a COE between 98 and 130 mills/kWh, direct capital cost between 4000 and 5000 M$, and Pe between 1000 and 1420 MW could be found with a minimum R0 between 5.1 and 6.7 m, and Pfusion between 2000 and 2650 MW

Introducing unstructured mesh support to a developing gyrokinetic code, gKPSP2

To solve governing differential equations over complex geometry, it is imperative to create and utilize unstructured mesh, in general. National Fusion Research Institute started developing a gyrokinetic simulation code, gKPSP2, to aim at a whole device modeling for real geometry of Tokamaks such as KSTAR, ITER and K-DEMO. Handling the entire domain of Tokamak is expected to require high performance parallel computation over more than 1M real space elements in addition to velocity space elements for each real space element. In this presentation, we introduce the code development in the mesh perspective and show how unstructured mesh is created for realistic Tokamak geometry and what functionalities are required to support the unstructured mesh in parallel use for gKPSP2. * This research was supported by R&D Program of "Study of an efficient SOL discretization algorithm for global ITER burning plasma simulation (code No. IN2004-6)" through the National Fusion Research Institute of Korea (NFRI) funded by the Government funds. Keyword : Unstructured mesh, Whole device modeling, Gyrokinetic code, High performance computin