We demonstrate the reduced computational cost of applying adaptive sparse grids to the problem of linear kinetic simulation of runaway electron dynamics. This work is motivated by the need to routinely perform high-dimensional kinetic simulation in a noise-free manner, with the implication being that a reduced computational cost for kinetic simulation allows for more routine investigation. Here we verify our solver with a series of tests (e.g., Fig. 1), including demonstrating that it yields the expected runaway electron production rate. We show that a non-adapted sparse grid approach yields significant savings in the computational resources required to resolve the runaway electron dynamics to sufficient accuracy. We also address a common criticism of sparse grids, in that we show that adding adaptivity to sparse grids recovers the advantage of sparse grids when the features of the solution are not coordinate aligned.
Our linear model includes terms for electric field acceleration, Coulomb collisions, and radiation damping. Figure 1 shows a test where we simulate on a sparse grid for a sufficiently small accelerating electric field that the distribution relaxes to a known Maxwellian (dotted line) in steady-state. There are several other efforts focused on the phase space simulation of runaway electron dynamics (e.g., 1, 2, , ). However, of these, it is only the Monte-Carlo approaches that have begun looking at including the full spatial variation required for quantitative analysis and prediction. This is because the Monte-Carlo methods require less computational resources than the continuum (mesh/grid-based) methods. As these approaches are advanced to include a self-consistent coupling of the distribution function evolution and field solutions to capture how the runaway population feeds back on the accelerating electric field, a continuum approach that can incorporate the spatial dimensions at reasonable compute cost is motivated. Here we investigate if adaptive sparse grids [5,6] allow for calculating the dynamics of the phase space distribution of electrons in a robust and accurate manner. Our approach is constructing a high-order, discontinuous Galerkin solver using a "sparse" polynomial space where the sparsity is introduced via a truncated tensor product of 1D bases. We employ upwinding for advection terms and LDG  for diffusion operators.
We further demonstrate (cf., Fig. 2) that adding adaptivity to the sparse grid discretization recovers the advantages of sparse grids in such where it is traditionally lost. The truncated tensor product drops, for example, those elements from the space whose total degree is less than the maximum degree in any one dimension. This will lead to cases where the advantages of the sparse grid are lost when a solution relies on the parts of the tensor product which were thrown away. Figure 2 demonstrates how adaptivity makes sparse grids robust to this issue.
In summary, the adaptive sparse grid approach appears to be a viable approach to enabling more routine kinetic simulation of fusion plasmas. We will continue to develop and apply this approach to incorporate more of the spatial dimensionality typically lacking from continuum simulation of runaway electron dynamics.
This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725.
1 D. del-Castillo-Negrete et al., Physics of Plasmas 25.5 (2018): 056104.
2 M. Landreman, et al., Comp. Physics Comm. 185.3 (2014): 847-855.
 R. W. Harvey et al., Physics of Plasmas 7.11 (2000): 4590-4599.
 D. Daniel et al., arXiv preprint arXiv:1902.10241 (2019).
 D. Pflüger et al., Journal of Complexity 26.5 (2010): 508-522.
 W. Guo et al., SIAM Journal on Scientific Computing 38.6 (2016): A3381-A3409.
 B. Cockburn et al., SIAM Journal on Numerical Analysis 35.6 (1998): 2440-2463.
|Affiliation||Oak Ridge National Laboratory|
|Country or International Organization||United States|