Fully Implicit Kinetic Solution of Collisional Plasmas

Abstract

In this paper we describe a numerical technique, matrix-free Newton–Krylov, for solving a single species, stationary, one spatial dimension, one velocity dimension, Vlasov–Fokker–Planck equation. This method is both deterministic and fully implicit and may not have been a viable option before recent developments in numerical methods. It is demonstrated that efficient steady-state solutions to the nonlinear integro-differential equation, can be achieved, obtaining quadratic convergence, but not incurring the large memory requirements of an integral operator. We present a model problem which simulates ion transport in the edge plasma of a tokamak fusion reactor and use this model problem to demonstrate the performance of the new solution method. We demonstrate that the solution algorithm is compatible with a higher-order, monotone, convective differencing scheme.