AI 中文总结
该研究针对多尺度线性动力学输运方程，提出一种保渐近动力学低秩半拉格朗日方法，结合半拉格朗日与低秩表示优势，经分析和数值实验验证其保渐近、大时间步长稳定且计算高效。
AI 中文摘要
本文针对多尺度线性动力学输运方程，开发了一种保渐近（AP）动力学低秩半拉格朗日方法。该方法结合了半拉格朗日离散化的大时间步长能力，以及低秩表示带来的存储与成本降低优势。所提格式将近似宏观密度更新与针对动力学分布的基更新伽辽金积分器耦合，为在半拉格朗日通量评估中保持降低的复杂度，通过采样角求积策略计算通量导数。我们在常系数情形下对全求积低秩格式建立了无条件稳定性分析，量化了通量导数中角采样引入的误差，证明所得格式在扩散极限下是保渐近的。包含高维测试案例的数值实验表明，所提方法是保渐近的，在大时间步长下稳定，且在动力学与扩散 regime 中计算效率高。
英文摘要
In this paper, we develop an asymptotic-preserving (AP) dynamical low-rank semi-Lagrangian method for multiscale linear kinetic transport equations. The method combines the large-time-step capability of semi-Lagrangian discretizations with the storage and cost reduction provided by low-rank representations. The proposed scheme couples an approximate macroscopic density update with the basis update Galerkin integrator for the kinetic distribution. To retain the reduced complexity in the semi-Lagrangian flux evaluation, the flux derivative is computed through a sampled angular quadrature strategy. We establish an unconditional stability analysis of the full-quadrature low-rank scheme in the constant-coefficient case. The error induced by angular sampling in the flux derivative is quantified. The resulting scheme is shown to be AP in the diffusive limit. Numerical experiments, including high-dimensional test cases, demonstrate that the proposed method is AP, stable under large time steps, and computationally efficient across kinetic and diffusive regimes.
Comments30 pages, 8 figures, 4 tables, 53 references
