Non-uniform splines for semi-Lagrangian kinetic simulations of the plasma sheath

Abstract

Numerical methods based on non-uniform splines of varying degrees are used to run kinetic semi-Lagrangian simulations of the plasma sheath. The sheath describes a region of plasma in contact with a wall, acting as a heat, momentum, and particle bath. The latter regulates the current loss and consequently the electric field. At micro scale it then influences the development of plasma turbulence and at a larger scale plasma rotation. However this region is particularly difficult to simulate, due to its kinetic nature and the presence of steep gradients requiring fine numerical resolution. In this work, a new well-conditioned method for determining the coefficients for Schoenberg's “best” quadrature is presented, providing improved precision compared to a naive approach based on b-spline derivatives (4 extra significant figures of precision are obtained for coefficients of the fifth order scheme). The construction of a simulation grid from Greville abcissae of judiciously chosen non-uniform knots is shown to reduce the memory requirements by 89% for the simulation presented. Cubic splines are found to provide a good compromise between fast convergence and controlled oscillations preserving positivity. Despite the reduction in performance associated with using non-uniform spline schemes, effective parallelisation through the use of GPUs leads to non-uniform simulations running 5.5 times faster than uniform simulations providing equivalent results.