A multigrid method for drift-kinetic calculations in stellarators and rippled tokamaks

Important phenomena such as the bootstrap current and collisional transport in stellarators, and neoclassical toroidal viscosity in tokamaks, must be computed by numerical solution of the drift-kinetic equation in nonaxisymmetric geometry. This equation has the form of a linear, inhomogeneous, advection-dominated advection-diffusion PDE with recirculating flow, internal boundary layers, and a null space, with typically five coupled dimensions (poloidal and toroidal angle, speed, velocity pitch angle, and species.) While multigrid algorithms are a preferred method for efficient solution of some PDEs, multigrid smoothers are typically unstable for accurate discretizations of the drift-kinetic equation due to the absence of any physical diffusion in the spatial dimensions, and the dominance of advection over diffusion in the velocity dimensions. In this work we demonstrate a high-order-accurate multigrid solution of the drift-kinetic equation in nonaxisymmetric geometry. A defect correction approach is used: solution of a high-order discretized problem is preconditioned by a multigrid cycle in which a low-order upwinded discretization is used for smoothing. Compared to a direct solver, the multigrid method can reduce the memory requirement by several orders of magnitude.