一种保持结构的惩罚方法用于单物种罗森布卢特-福克-普朗克方程
A Structure-Preserving Penalization Method for the Single-species Rosenbluth-Fokker-Planck Equation
AI总结：
本文提出了一种结构保持的惩罚方法，用于解决单物种罗森布卢特-福克-普朗克方程的刚性问题，通过三种新方法保证了守恒性和正性。
摘要：
罗森布卢特-福克-普朗克（RFP）方程描述等离子体中及跨物种的库仑碰撞动力学。它属于更广泛的各向各异性扩散-对流方程类，其数值近似极具挑战性，因为其非线性、刚性和结构性质如守恒和熵耗散（因此最大布满分布为平衡状态）。在本文中，我们提出了一种保持结构的惩罚方案，用于刚性的单物种RFP方程。该方案具有三个新颖的组成部分：1）一种新的罗森布卢特-福克-普朗克方程的已知切变-合作离散化的泛化，该方法保持平衡并能够在保持质量、动量和能量的同时保持正性；2）一种易于逆变的各向同性变量系数惩罚操作符，用于处理时间刚性而无需使用完全隐式方案，借鉴了显式-隐式-空（EIN）方法的思想；3）一种自适应时间步长策略，该策略保持了完整惩罚方案的正性。所得到的方案严格保持质量、动量和能量，是无条件稳定的，并且稳健地保持正性。该方案通过线性和非线性各向各异性扩散示例进行了演示，包括多个单物种RFP示例。
英文摘要：
The Rosenbluth-Fokker-Planck (RFP) equation describes Coulomb collisional dynamics within and across species in plasmas. It belongs to the broader class of anisotropic-diffusion-advection equations, whose numerical approximation is highly-nontrivial due to its nonlinearity, stiffness, and structural properties such as conservation and entropy dissipation (hence with the Maxwellian distribution as the equilibrium state). In this paper, we propose a structure-preserving penalization scheme for the stiff, single-species RFP equation. The scheme features three novel components: 1) a novel generalization of the well-known Chang-Cooper discretization for the RFP equation that is equilibrium-preserving and enables positivity while preserving mass, momentum, and energy; 2) an easy-to-invert isotropic variable-coefficient penalization operator to deal with the temporal stiffness without resorting to a fully implicit scheme, borrowing ideas from explicit-implicit-null (EIN) methods, and 3) an adaptive timestepping strategy that preserves the positivity of the full penalized scheme. The resulting scheme conserves mass, momentum, and energy strictly, is unconditionally stable, and robustly positivity preserving. The scheme is demonstrated with linear and nonlinear anisotropic diffusion examples of increasing complexity, including several single-species RFP examples.
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