For more than a decade, CASC researchers have been collaborating with the LLNL Fusion Energy Program (FEP) in the development of advanced numerical methodologies for magnetic fusion energy (MFE) applications. The LLNL FEP is a recognized world leader in MFE research, especially in research focused on the “edge” region of tokamak fusion reactors, where the understanding of plasma behavior is critical for reactor design and performance optimization. As part of a project jointly funded by the DOE Offices of Fusion Energy Sciences (FES) and Advanced Scientific Computing Research (ASCR), CASC and FEP researchers are developing an edge plasma simulation code named COGENT that employs new computational techniques.
First among the simulation challenges is the fact that edge plasmas cannot be modeled well as fluids. The use of approximate gyrokinetic models provides some simplification of an otherwise fully kinetic model, but one is still faced with the need to solve a system of partial differential equations in five independent variables plus time, in addition to some form of Maxwell’s equations. Other numerical challenges result from the strong anisotropy associated with plasma flow along magnetic field lines, geometric constraints imposed by the sheared magnetic geometry comprised of open and closed field lines, and simultaneous treatment of multiple time and spatial scales corresponding to plasma (micro-scale) turbulence and (macro-scale) transport.
Although particle-based approaches provide a natural choice for kinetic simulations, the associated noise and accuracy uncertainties pose a significant concern. As an alternative, COGENT employs a continuum (Eulerian) approach in which the gyrokinetic system is discretized and solved on a phase space grid. The computational costs associated with the use of grids in four (assuming axisymmetric configuration space geometry) or five spatial dimensions are partially mitigated via fourth-order discretization techniques, which reduce the number of grid cells needed to achieve a specific accuracy relative to lower-order methods. The edge geometry is described using a multiblock approach in which subregions are mapped to locally rectangular grids that are further domain decomposed for parallelization. Alignment of the block mappings with the magnetic field and a specially-developed discretization of the phase space velocities isolates the strong flow along field lines for improved discretization accuracy. Multiple time scales are treated using an Additive Runge-Kutta (ARK) time integration framework in which the CASC-developed Hypre BoomerAMG algebraic multigrid solver is employed in preconditioners for the Newton-Krylov nonlinear solves required in the implicit ARK stages.