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

TL;DR

This work introduces a noise-robust Monte Carlo Gradient Approximation (MCGA) to directly evolve the electric field $\mathbf{E}$ in 2D, bypassing noisy finite-difference gradients of the electric potential $\varphi$ used in EMC3. By deriving the evolution equation $\partial_t \mathbf{E} = \nabla\cdot\left(D(\mathbf{r})\nabla \mathbf{E}\right) + \nabla\cdot\left(\nabla D(\mathbf{r}) \otimes \mathbf{E}\right)$ and decomposing into $E_x$ and $E_y$, the method enables Monte Carlo estimation of the electric-field gradient with reduced noise amplification, at the cost of neglecting a cross-component source term $\partial_y E_y$ in practice. Numerical tests using manufactured solutions demonstrate that MCGA has lower relative error than finite-difference approaches and exhibits superior noise robustness when grid resolution is refined. The approach advances EMC3 physics by enabling more reliable self-consistent $E\times B$ drift modeling, with potential extensions to Bohm boundary conditions and non-rectangular grids in future work.

Abstract

One of the main codes to analyze and optimize stellarator configurations is the EMC3 code, which implements a state-of-the-art 3D Monte Carlo plasma edge transport code. However, so far, a self-consistent treatment of the E x B drift is absent. This plasma drift is known to significantly impact the particle and heat distribution in the plasma edge. It is desirable to incorporate this drift into EMC3 to improve the predictive capabilities of the code. The calculation of the E x B drift requires the approximation of the electric field E, which is proportional to the gradient of the electric potential $ \varphi $. In previous work, the gradient was calculated with a least squares method based on a finite difference approximation of the electric potential. However, due to the stochastic nature of EMC3, the output plasma fields computed by the code are inherently noisy. The finite difference method further amplifies the noise, with the amplification growing as the grid size decreases. We continue from, which introduced a new noise-robust method for 1D derivatives. We extend the noise-robust method to 2D and apply it to the electric potential. We show that a PDE can be derived that describes the evolution of the electric field in case of a uniform diffusion coefficient. This PDE allows us to approximate the electric field directly with a Monte Carlo simulation, thus avoiding the need for a finite difference approximation. We illustrate the accuracy of the method and the noise robustness with a test case.

Figures (3)

• Figure 1: Comparison of the MCGA and finite difference approximation of the electric field $\bm{E}$ and the norm $\|\bm{E} \|_{2}$. For the Monte Carlo method, two cases are considered: one where the term $\partial_{y} E_y$ in Eq. \ref{['eq:experiment1:source_x']} is set to zero, and another where it is given by the exact source term.
• Figure 2: Comparison of the relative error at each grid point for the MCGA and FD methods. As before, the MCGA method is shown both with $\partial_{y}E_{y}$ set to zero and with its exact value. Note that the colorbar scales are different between the plots.
• Figure 3: Effect of the grid resolution on the variance. The variance of the FD approximation of $E_{x}$ grows faster than the MC approximation of $E_{x}$. This indicates that the MC method is more robust to noise increases in noise as the grid size decreases.