# Probabilistic schemes for semi-linear nonlocal diffusion equations with application in predicting runaway electron dynamics

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## Date

2019-12-05## Type of Degree

PhD Dissertation## Department

Mathematics and Statistics

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Show full item record## Abstract

In this work, we focus on developing and analyzing novel probabilistic numerical approaches for solving several types of semi-linear nonlocal diffusion equations in both unbounded and bounded high dimensional spaces. First, we propose probabilistic schemes for solving the partial integro-differential equation (PIDE) with Fokker-Planck operator related to a jump-diffusion process in the unbounded domain $\mathbb{R}^d$. Under a given probability space, we exploit the probabilistic representation of the solution of PIDE to construct both temporal discrete and temporal-spatial discrete schemes of the solution of PIDE. The rigorous error analysis is provided to prove that the temporal discrete scheme can achieve first-order convergence. To add in spatial discretization, the temporal-spatial discrete scheme incorporates with the high-order piecewise polynomial interpolation that leads to high order convergence with respect to spatial mesh size $\Delta x$. Next, we consider another typical nonlocal diffusion equation, the fractional Laplacian equations. Due to work studied by Serge Cohen and Jan Rosi\'eski (2007), and S. Asmussen and Jan Rosi\'eski (2001), stable processes can be simulated by L\'evy processes that consist of the compound Poisson processes and appropriate Brownian motions. Hence the fractional Laplacian operator can be approximated by the second partial integro-differential operator. Our probabilistic schemes for the PIDE model introduced in Chapter \ref{pide_ch} can give a novel numerical approach for solving the fractional Laplacian equation. Third, we impose volume constraints into PIDEs model and consider the initial-boundary value partial integro-differential equations. The key idea is to exploit the regularity of the solution $u(t,x)$ to avoid direct approximation of the random exit time corresponding to the boundary conditions. The error from the exit time decays sub-exponentially with respect to temporal mesh $\Delta t$ when all interior grid points are sufficiently far from the boundary. Moreover, our numerical methods lead to an overall first-order convergence rate with respect to $\Delta t$ and achieve high order convergence with respect to $\Delta x$. Last, we introduce one application of the initial-boundary value PIDE problem, the approximation of the runaway probability of electrons in fusion tokamak simulation. Runaway electrons (REs) generated during magnetic disruptions present a major threat to the safe operation of fusion tokamak. A critical aspect of understanding REs dynamics is to calculate runaway probabilities, i.e., the probability of an electron in the phase space will runaway on, or before, a time $t>0$. Mathematically, such probability can be obtained by solving the adjoint equation of the underlying Fokker-Planck equation that controls the electron dynamics. In this work, we present a sparse-grid probabilistic scheme for computing runaway probability. The key ingredient of our approach is to represent the solution of the adjoint equation as a conditional expectation, such that discretizing the differential operator becomes approximating a set of integrals. The sparse grid interpolation is utilized to approximate the runaway probability, and adaptive refinement is also exploited to handle the sharp transition layer between the runaway and non-runaway region.