碰撞等离子体模型的渐近保持确定性粒子方法
Asymptotic-preserving deterministic particle methods for collisional plasma models
AI总结：
本文提出了一种新型渐近保持确定性粒子方法，用于碰撞等离子体模型，包括Landau-Fokker-Planck和Dougherty碰撞算子，在流体尺度下。方法通过能量守恒的JKO方案处理非刚性和刚性碰撞算子，展示了结构保持和渐近保持特性。
摘要：
我们开发了用于碰撞等离子体模型的新渐近保持（AP）确定性粒子方法，包括Landau-Fokker-Planck和Dougherty碰撞算子，在流体尺度下。我们的方案通过利用其梯度流结构，通过能量守恒的Jordan-Kinderlehrer-Otto（JKO）方案，将非刚性传输部分显式处理，刚性碰撞算子完全隐式处理。这种方法扩展了我们之前关于空间均匀Landau方程的工作[arXiv:2409.12296]，并引入了通过投影梯度流公式处理Dougherty算子的新方法。我们识别了Jacobian对数行列式评估在刚性区域中的关键作用，并引入了内时间四阶策略以提高准确性和效率。此外，我们揭示了与基于分数的传输建模的有趣联系，表明显式和隐式分数匹配作为我们统一变分框架的特殊情况，并在刚性区域中表现出限制。我们还通过神经网络参数化和高效训练策略开发了实用的大规模实现。各种数值示例展示了我们的方案在一般初始数据下的结构保持和AP特性。
英文摘要：
We develop novel asymptotic-preserving (AP) deterministic particle methods for collisional plasma models, including both Landau--Fokker--Planck and Dougherty collision operators, under hydrodynamic scaling. Our schemes treat the non-stiff transport part explicitly and the stiff collision operators fully implicitly through the energy-conserving Jordan--Kinderlehrer--Otto (JKO) schemes by exploiting their gradient flow structures. This approach extends our previous work on the space-homogeneous Landau equation [arXiv:2409.12296] and introduces a new treatment of the Dougherty operator via a projected gradient flow formulation. We identify the crucial role of Jacobian log-determinant evaluation in stiff regimes and introduce an inner-time quadrature strategy that improves both accuracy and efficiency. Furthermore, we uncover intriguing connections with score-based transport modeling, showing that both explicit and implicit score matching arise as special cases of our unified variational framework and exhibit limitations in the stiff regime. We also develop practical large-scale implementations via neural network parameterization and efficient training strategies. Various numerical examples demonstrate the structure-preserving and AP properties of our schemes for general initial data.
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