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.

中文翻译：

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用于失控电子的完全隐式、可扩展、保守的非线性相对论 Fokker-Planck 0D-2P 求解器

在施加足够强的电场后，电子脱离热平衡并接近相对论速度。这些高能失控电子 (MeV) 在托卡马克破坏物理学中发挥着重要作用，因此准确理解它们对于制定可靠的缓解策略至关重要。为此，我们为 0D-2P（即，包括两个动量坐标）相对论非线性 Fokker-Planck 方程 (rFP) 开发了一个完全隐式求解器。与早期的隐式 rFP 研究（NORSE、CQL3D）一样，电子-离子相互作用使用洛伦兹算子建模，同步加速器阻尼使用亚伯拉罕-洛伦兹-狄拉克反应项建模。然而，我们的实施通过以下方式改进了这些早期研究：1）确保电子碰撞的精确守恒特性，2) 严格保持积极性，以及 3) 在算法上和并行上可扩展。我们提出的方法的关键是线性化 rFP 方程的高效多重网格预处理器，Braams-Karney 势的多重网格椭圆求解器 [Braams and Karney, Phys. 牧师莱特。59, 16 (1987)]，以及一种确定相关边界值的新型自适应技术。我们通过数值结果验证了所提出方案的准确性和效率，从小电场电导率测量到施加强电场时失控尾部动力学的准确再现。Braams-Karney 势的多重网格椭圆求解器 [Braams and Karney, Phys. 牧师莱特。59, 16 (1987)]，以及一种确定相关边界值的新型自适应技术。我们通过数值结果验证了所提出方案的准确性和效率，从小电场电导率测量到施加强电场时失控尾部动力学的准确再现。Braams-Karney 势的多重网格椭圆求解器 [Braams and Karney, Phys. 牧师莱特。59, 16 (1987)]，以及一种确定相关边界值的新型自适应技术。我们通过数值结果验证了所提出方案的准确性和效率，从小电场电导率测量到施加强电场时失控尾部动力学的准确再现。