SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 73-92, February 2024.
Abstract. Radiation transport problems are posed in a high-dimensional phase space, limiting the use of finely resolved numerical simulations. An emerging tool to efficiently reduce computational costs and memory footprint in such settings is dynamical low-rank approximation (DLRA). Despite its efficiency, numerical methods for DLRA need to be carefully constructed to guarantee stability while preserving crucial properties of the original problem. Important physical effects that one likes to preserve with DLRA include capturing the diffusion limit in the high-scattering regimes as well as dissipating energy. In this work we propose and analyze a dynamical low-rank method based on the “unconventional” basis update & Galerkin step integrator. We show that this method is asymptotic preserving, i.e., it captures the diffusion limit, and energy stable under a CFL condition. The derived CFL condition captures the transition from the hyperbolic to the parabolic regime when approaching the diffusion limit.

中文翻译：

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渐近守恒且能量稳定的动态低阶逼近

SIAM 数值分析杂志，第 62 卷，第 1 期，第 73-92 页，2024 年 2 月。
摘要。辐射传输问题是在高维相空间中提出的，限制了精细解析数值模拟的使用。在这种情况下，一种可以有效降低计算成本和内存占用的新兴工具是动态低秩近似（DLRA）。尽管其效率很高，但 DLRA 的数值方法需要仔细构建，以保证稳定性，同时保留原始问题的关键属性。人们喜欢用 DLRA 保留的重要物理效应包括捕获高散射区域中的扩散极限以及耗散能量。在这项工作中，我们提出并分析了一种基于“非常规”基更新和伽辽金步积分器的动态低秩方法。我们证明该方法是渐近保持的，即它捕获了扩散极限，并且在 CFL 条件下能量稳定。导出的 CFL 条件捕获了接近扩散极限时从双曲线到抛物线状态的转变。