On the stability of robust dynamical low-rank approximations for hyperbolic problems
On the stability of robust dynamical low-rank approximations for hyperbolic problems
复制标题
双曲问题鲁棒动态低阶近似的稳定性
DOI：
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发表时间：
2021
影响因子：
3.1
通讯作者：
Gianluca Ceruti
中科院分区：
文献类型：
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作者：
J. Kusch;L. Einkemmer;Gianluca Ceruti
动态低秩近似（DLRA）是用来处理高维问题，出现在这样的不同领域的动力学输运和不确定性量化。尽管众所周知，某些空间和时间离散化与DLRA方法相结合时会导致数值不稳定性，但人们对这种现象的理解很少。在本文中，我们进行了相应的非线性运动方程的L2稳定性分析。这揭示了投影分裂积分器的不稳定性的来源时，首先离散方程，然后应用DLRA。在此基础上，我们提出了一个投影分裂积分器，应用DLRA连续系统进行离散化之前，恢复经典CFL条件。我们还表明，非常规的积分器具有更有利的稳定性，并解释了为什么投影分裂积分器表现更好时，近似较高的时刻，而非常规的积分器一般是上级的一阶矩。此外，提出了一个有效的和稳定的动力学输运中的散射项的动力学低秩更新。动力学传输和不确定性量化，这证实了稳定性分析的结果，数值实验。
The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform a L2 stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector splitting integrator when first discretizing the equations and then applying the DLRA. Based on this we propose a projector splitting integrator, based on applying DLRA to the continuous system before performing the discretization, that recovers the classic CFL condition. We also show that the unconventional integrator has more favorable stability properties and explain why the projector splitting integrator performs better when approximating higher moments, while the unconventional integrator is generally superior for first order moments. Furthermore, an efficient and stable dynamical low-rank update for the scattering term in kinetic transport is proposed. Numerical experiments for kinetic transport and uncertainty quantification, which confirm the results of the stability analysis, are presented.
使用人工散射的离散坐标法的射线效应缓解
影响因子：
1.2
作者：
Frank, Martin;Kusch, Jonas;Camminady, Thomas;Hauck, Cory D.
通讯作者：
Hauck, Cory D.
影响因子：
1.2
作者：
Stephan Gerster;M. Herty
通讯作者：
M. Herty
线性玻尔兹曼方程的动态低阶积分器：扩散极限中的误差分析
影响因子：
2.9
作者：
Ding, Zhiyan;Einkemmer, Lukas;Li, Qin
通讯作者：
Li, Qin