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 \(\mathcal {O}(\varDelta t)\) 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.
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Notes
- 1.
ITER (originally the International Thermonuclear Experimental Reactor) is an international nuclear fusion research and engineering mega project, which will be the world’s largest magnetic confinement plasma physics experiment. See https://www.iter.org/ for details.
- 2.
- 3.
The escape time \(\tau _{t_n,x}\) in Eq. (3.6) should be defined by replacing \(\boldsymbol X_s^{t,x}\) with the Euler discretization, i.e., \(\boldsymbol X_{s}^{t_n,x} = \boldsymbol x + b(\boldsymbol x)(s-t_n)+ \sigma (\boldsymbol x)(\boldsymbol W_s-\boldsymbol W_{t_n})\) for s ≥ t n in Eq. (3.2). We use the same notation without creating confusion.
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Acknowledgements
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.
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Yang, M., Zhang, G., del-Castillo-Negrete, D., Stoyanov, M., Beidler, M. (2021). A Sparse-Grid Probabilistic Scheme for Approximation of the Runaway Probability of Electrons in Fusion Tokamak Simulation. In: Bungartz, HJ., Garcke, J., Pflüger, D. (eds) Sparse Grids and Applications - Munich 2018. Lecture Notes in Computational Science and Engineering, vol 144. Springer, Cham. https://doi.org/10.1007/978-3-030-81362-8_11
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