A Sparse-Grid Probabilistic Scheme for Approximation of the Runaway Probability of Electrons in Fusion Tokamak Simulation

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Runaway electrons (RE) generated during magnetic disruptions present a major threat to the safe operation of plasma nuclear fusion reactors. A critical aspect of understanding RE dynamics is to calculate the runaway probability, i.e., the probability that an electron in the phase space will runaway on, or before, a prescribed time. Such probability can be obtained by solving the adjoint equation of the underlying Fokker-Planck equation that controls the electron dynamics. In this effort, we present a sparse-grid probabilistic scheme for computing the 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 reduces to the approximation of a set of integrals. Adaptive sparse grid interpolation is utilized to approximate the map from the phase space to the runaway probability. The main novelties of this effort are the integration of the sparse-grid method into the probabilistic numerical scheme for computing escape probability, and the application of the proposed method in computing RE probabilities. Two numerical examples are given to illustrate that the proposed method can achieve O(Δt) convergence, and that the local anisotropic adaptive refinement strategy (M. Stoyanov, Adaptive sparse grid construction in a context of local anisotropy and multiple hierarchical parents. In: Sparse Grids and Applications-Miami 2016, Springer, Berlin, 2018, pp. 175–199) can effectively handle the sharp transition layer between the runaway and non-runaway regions.

Original languageEnglish
Title of host publicationSparse Grids and Applications - 2018
EditorsHans-Joachim Bungartz, Jochen Garcke, Jochen Garcke, Dirk Pflüger
PublisherSpringer Science and Business Media Deutschland GmbH
Pages245-264
Number of pages20
ISBN (Print)9783030813611
DOIs
StatePublished - 2021
Event5th Workshop on Sparse Grids and Applications, SGA 2018 - Munich, Germany
Duration: Jul 23 2018Jul 27 2018

Publication series

NameLecture Notes in Computational Science and Engineering
Volume144
ISSN (Print)1439-7358
ISSN (Electronic)2197-7100

Conference

Conference5th Workshop on Sparse Grids and Applications, SGA 2018
Country/TerritoryGermany
CityMunich
Period07/23/1807/27/18

Funding

Acknowledgments This material is based upon work supported in part by the U.S. Department of Energy, Office of Science, Offices of Advanced Scientific Computing Research and Fusion Energy Science, and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC, for the U.S. Department of Energy under Contract DE-AC05-00OR22725.

Keywords

  • Adjoint equation
  • Feynman-Kac formula
  • Fokker-Planck equation
  • Runaway electrons
  • Runaway probability
  • Sparse grids

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