A probabilistic approach for the computation of confinement and exit-time in local and non-local plasma transport problems

The exit time probability, which gives the likelihood that a particle leaves a prescribed region in the phase space of a dynamical system at, or before, a given time, is arguably one of the most natural and important transport problems in general and in magnetically confined plasmas in particular. This talk presents a novel numerical method for computing this probability, and the confinement statistics, for local and nonlocal plasma transport problems described by time-dependent Fokker-Planck (FP) differential and integro-differential equations, and their equivalent stochastic differential equations (SDEs) [1,2]. The method is based on the direct numerical evaluation of the Feynman-Kac formula that establishes a link between the adjoint FP equation and the forward SDE. In the local transport case, the SDEs are driven by Brownian motion, and in the nonlocal case, by Poisson jump processes describing nonlocality with a finite horizon kernel in the corresponding integro-differential FP equation. The efficiency and accuracy of the proposed method rests on the reduction of the computational complexity of the problem to the evaluation of Gaussian quadratures. The method does not face the noise limitations of direct Monte-Carlo algorithms, and bypasses stability and efficiency issues of standard finite-difference methods for time-dependent FP equations.  In particular, the proposed method is unconditionally stable, exhibits second-order convergence in space, first-order convergence in time, and it is straightforward to parallelize. Several applications are presented, including the production of runaway electrons in tokamak disruptions, ExB transport, and non-local transport resulting from Landau-fluid type kinetic closures.

[1]  https://arxiv.org/abs/2104.14561 Accepted for  publication in Journal of Computational Physics (2021).

[2]  https://arxiv.org/abs/2001.05800 Springer Verlag Lecture Notes (2021).