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.

翻译：本文研究了一种包含逃逸电子相对论漂移动力学模型、小角度库仑碰撞Fokker-Planck算子、辐射阻尼算子以及次级敲出（Boltzmann）碰撞源的模型。我们开发了一种基于有限体积法和守恒有限差分格式的新型可扩展全隐式求解器，并引入动态网格自适应技术。基于p4est库，我们在PETSc库中构建了新型数据管理框架，实现了动态自适应网格细化（AMR）、分布式内存并行化以及计算任务的动态负载均衡。该框架及其上建立的逃逸电子求解器能够动态捕捉低能区域的体麦克斯韦分布与高能区域的逃逸尾分布。为通过AMR算法有效捕捉特征，我们提出了一种伴随隐式时间演化的新型AMR指示器预测策略。该策略辅以计算成本低廉的基于特征的AMR指示器，并从理论上对其进行分析。数值结果量化了该预测策略在特征捕捉方面相较于非预测策略的优势，同时展示了计算成本间的权衡关系。通过包括制造解和不同物理模型解在内的多个基准问题，我们验证了模型参数的鲁棒性、算法可扩展性及并行扩展性，重点展示了隐式时间步进与AMR在逃逸电子模拟中的优势。