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Scalable Implicit Solvers with Dynamic Mesh Adaptation for a Relativistic Drift-Kinetic Fokker-Planck-Boltzmann Model

Johann Rudi, Max Heldman, Emil M. Constantinescu, Qi Tang, Xian-Zhu Tang

TL;DR

A relativistic drift-kinetic model for runaway electrons along with a Fokker-Planck operator for small-angle Coulomb collisions, a radiation damping operator, and a secondary knock-on (Boltzmann) collision source is considered and a new scalable fully implicit solver is developed.

Abstract

In this work we consider a relativistic drift-kinetic model for runaway electrons along with a Fokker-Planck operator for small-angle Coulomb collisions, a radiation damping operator, and a secondary knock-on (Boltzmann) collision source. We develop a new scalable fully implicit solver utilizing finite volume and conservative finite difference schemes and dynamic mesh adaptivity. A new data management framework in the PETSc library based on the p4est library is developed to enable simulations with dynamic adaptive mesh refinement (AMR), distributed memory parallelization, and dynamic load balancing of computational work. This framework and the runaway electron solver building on the framework are able to dynamically capture both bulk Maxwellian at the low-energy region and a runaway tail at the high-energy region. To effectively capture features via the AMR algorithm, a new AMR indicator prediction strategy is proposed that is performed alongside the implicit time evolution of the solution. This strategy is complemented by the introduction of computationally cheap feature-based AMR indicators that are analyzed theoretically. Numerical results quantify the advantages of the prediction strategy in better capturing features compared with nonpredictive strategies; and we demonstrate trade-offs regarding computational costs. The robustness with respect to model parameters, algorithmic scalability, and parallel scalability are demonstrated through several benchmark problems including manufactured solutions and solutions of different physics models. We focus on demonstrating the advantages of using implicit time stepping and AMR for runaway electron simulations.

Scalable Implicit Solvers with Dynamic Mesh Adaptation for a Relativistic Drift-Kinetic Fokker-Planck-Boltzmann Model

TL;DR

A relativistic drift-kinetic model for runaway electrons along with a Fokker-Planck operator for small-angle Coulomb collisions, a radiation damping operator, and a secondary knock-on (Boltzmann) collision source is considered and a new scalable fully implicit solver is developed.

Abstract

In this work we consider a relativistic drift-kinetic model for runaway electrons along with a Fokker-Planck operator for small-angle Coulomb collisions, a radiation damping operator, and a secondary knock-on (Boltzmann) collision source. We develop a new scalable fully implicit solver utilizing finite volume and conservative finite difference schemes and dynamic mesh adaptivity. A new data management framework in the PETSc library based on the p4est library is developed to enable simulations with dynamic adaptive mesh refinement (AMR), distributed memory parallelization, and dynamic load balancing of computational work. This framework and the runaway electron solver building on the framework are able to dynamically capture both bulk Maxwellian at the low-energy region and a runaway tail at the high-energy region. To effectively capture features via the AMR algorithm, a new AMR indicator prediction strategy is proposed that is performed alongside the implicit time evolution of the solution. This strategy is complemented by the introduction of computationally cheap feature-based AMR indicators that are analyzed theoretically. Numerical results quantify the advantages of the prediction strategy in better capturing features compared with nonpredictive strategies; and we demonstrate trade-offs regarding computational costs. The robustness with respect to model parameters, algorithmic scalability, and parallel scalability are demonstrated through several benchmark problems including manufactured solutions and solutions of different physics models. We focus on demonstrating the advantages of using implicit time stepping and AMR for runaway electron simulations.
Paper Structure (29 sections, 1 theorem, 47 equations, 15 figures, 7 tables, 1 algorithm)

This paper contains 29 sections, 1 theorem, 47 equations, 15 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Coulomb collision
  4. Synchrotron radiation
  5. Knock-on collision
  6. Computational domain and boundary conditions
  7. Conservation and an alternative RFP form
  8. Adaptive mesh refinement: Indicators and their prediction in time
  9. AMR indicators for coarsening and refinement
  10. Dynamic AMR and indicator prediction in time
  11. Relavitistic electron drift-kinetic solver based on PETSc and p4est
  12. DMBF: New PETSc data management for p4est-based AMR solvers
  13. Relativistic electron drift-kinetic solver based on DMBF
  14. Experimental setup
  15. Manufactured solution
  16. ...and 14 more sections

Key Result

Proposition 3.3

Given a Lipschitz continuous function $f>0$, let ${\chi[t]{\mathstrut}}_{LGS}$ be the gradient-scale AMR indicator defined in eq:lgs and ${\chi[t]{\mathstrut}}_{LDR}$ be the log-DR indicator defined in eq:ldr. Then the two indicators satisfy

Figures (15)

  • Figure 1: Diagram of dynamic AMR without prediction. In the "interp." (interpolation) step, the indicator ${\chi[t]{\mathstrut}}$ is computed from $f$, and this indicator is used to adapt the mesh (e.g., from mesh $A$ to mesh $B$). The "evolve" step advances $f$ forward in time by $\Delta t_\mathrm{adapt} = \Delta t$.
  • Figure 3: A mesh cell (blue square) is subdivided into four cells with one cell-centered DOF (back filled circle). The guard layer of this mesh cell is comprised of four adjacent mesh cells (brown squares), if the neighboring cells have the same level of refinement. Filling the guard layer's DOFs (black unfilled circles) of the (blue) mesh cell is done by copying the DOFs from adjacent cells (brown filled circles) in the case of uniform level of refinement.
  • Figure 4: Filling a guard layer's DOFs (black unfilled circles) in the case of adaptive refinement requires interpolating the DOFs of the adjacent mesh cell(s) (brown filled circles) to the locations at the guard layer's DOFs, while the latter are positioned to resemble a stretched, uniformly refined grid for the DOF of the mesh cell(s) (black filled circles).
  • Figure 5: Spatial mean of the AMR indicator LogDR \ref{['eq:ldr']} evolving in time: AMR without prediction (top) and AMR with prediction (bottom). Decreasing refinement frequencies (RF32, RF16, RF8, RF4) are lowering the LogDR metric, whereas with prediction the metric remains at low levels for any refinement frequency.
  • Figure 7: Spatial distribution of AMR indicator around the mean from Figure \ref{['fig:pred0-vs-pred1-mean']} that is shown as an envelope of one standard deviation above and below the mean curve. (Note: The vertical axis is clipped at zero because the AMR indicator is nonnegative.) AMR without prediction is shown in the top and AMR with prediction in the bottom graph.
  • ...and 10 more figures

Theorems & Definitions (6)

  • Definition 3.1: Gradient-scale indicator
  • Definition 3.2: Log-DR indicator
  • Proposition 3.3: Equivalence between gradient scale and log-DR indicators
  • proof
  • Remark 3.4
  • Remark 3.5