# A fully implicit, scalable, conservative nonlinear relativistic Fokker-Planck 0D-2P solver for runaway electron

## TL;DR

This paper introduces a fully implicit, scalable, and conservative solver for the relativistic nonlinear Fokker-Planck equation, improving accuracy and efficiency in modeling runaway electrons in tokamak disruptions.

## Contribution

It develops a novel implicit solver with exact conservation, positivity preservation, and scalable parallel algorithms for the relativistic Fokker-Planck equation including advanced multigrid techniques.

## Key findings

- Accurately reproduces runaway tail dynamics under strong electric fields.
- Ensures exact conservation and positivity in numerical solutions.
- Demonstrates efficiency and scalability through numerical tests.

## Abstract

Upon application of a sufficiently strong electric field, electrons break away from thermal equilibrium and approach relativistic speeds. These highly energetic runaway electrons (MeV) play a significant role in tokamak disruption physics, and therefore their accurate understanding is essential to develop reliable mitigation strategies. For this purpose, we have developed a fully implicit solver for the 0D-2P (i.e., including two momenta coordinates) relativistic nonlinear Fokker-Planck equation (rFP). As in earlier implicit rFP studies (NORSE, CQL3D), electron-ion interactions are modeled using the Lorentz operator, and synchrotron damping using the Abraham-Lorentz-Dirac reaction term. However, our implementation improves on these earlier studies by 1) ensuring exact conservation properties for electron collisions, 2) strictly preserving positivity, and 3) being scalable algorithmically and in parallel. Key to our proposed approach is an efficient multigrid preconditioner for the linearized rFP equation, a multigrid elliptic solver for the Braams-Karney potentials [Braams and Karney, Phys. Rev. Lett. 59, 16 (1987)], and a novel adaptive technique to determine the associated boundary values. We verify the accuracy and efficiency of the proposed scheme with numerical results ranging from small electric-field electrical conductivity measurements to the accurate reproduction of runaway tail dynamics when strong electric fields are applied.