A noise-robust Monte Carlo method for electric field calculations in EMC3

Abstract

EMC3 is a state-of-the-art 3D Monte Carlo code for plasma edge transport in stellarator configurations, but it does not yet treat the $E \times B$ drift self-consistently. Computing the drift requires the electric field $E = -\nabla \varphi$, and hence an accurate gradient of the electric potential $\varphi$. Because the plasma fields produced by EMC3 are inherently noisy, the finite difference approximation used previously amplifies this noise, increasingly so as the grid is refined. We extend the Monte Carlo Gradient Approximation method, originally developed for 1D Fokker--Planck equations, to a 2D setting and apply it to the electric potential. For an isotropic diffusion coefficient, we derive a PDE governing the evolution of the electric field itself, which allows $E$ to be approximated directly by a Monte Carlo simulation, avoiding finite differences altogether. A numerical experiment based on manufactured solutions demonstrates the accuracy of the method and shows that the variance grows substantially more slowly under grid refinement than for finite differences. The resulting formulation contains a source term involving the second poloidal derivative of the potential, which we neglect. This is admissible only when the radial scale of the plasma edge is thin compared to the poloidal variation scale, an ordering expected to fail near X-points, during detachment, and in island divertors such as W7-X. The present work should therefore be regarded as a proof of concept rather than a general electric field solver for EMC3.