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), parallel computation, and load balancing. This framework is tested through the development of the runaway electron solver that is 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 full solver is further verified through several benchmark problems including manufactured solutions and solutions of physics models. We particularly focus on demonstrating the advantages of using implicit time stepping and AMR for runaway electron simulations.

翻译：本文研究了一个针对逃逸电子的相对论性漂移动理学模型，该模型包含小角度库仑碰撞的Fokker-Planck算子、辐射阻尼算子以及次级碰撞（Boltzmann）源项。我们开发了一种新的可扩展全隐式求解器，采用有限体积法和保守有限差分格式，并结合动态网格自适应技术。基于p4est库，我们在PETSc库中构建了新的数据管理框架，以实现动态自适应网格细化（AMR）、并行计算和负载均衡。该框架通过开发逃逸电子求解器进行了测试，该求解器能动态捕获低能区域的体麦克斯韦分布和高能区域的逃逸尾迹。为通过AMR算法有效捕获特征，我们提出了一种新的AMR指标预测策略，该策略与解的隐式时间推进同步执行。该策略辅以计算成本低廉的基于特征的AMR指标，并从理论上进行了分析。数值结果量化了预测策略相比非预测策略在特征捕获方面的优势，同时展示了计算成本上的权衡。通过包括制造解和物理模型解在内的多个基准问题，进一步验证了完整求解器的性能。我们特别着重展示了在逃逸电子模拟中使用隐式时间步进和AMR的优势。