Dynamical low-rank approximation, as has been demonstrated recently, can be extremely efficient in solving kinetic equations. However, a major deficiency is that they do not preserve the structure of the underlying physical problem. For example, the classic dynamical low-rank methods violate mass, momentum, and energy conservation. In [L. Einkemmer, I. Joseph, J. Comput. Phys. 443:110495, 2021] a conservative dynamical low-rank approach has been proposed. However, directly integrating the resulting equations of motion, similar to the classic dynamical low-rank approach, results in an ill-posed scheme. In this work we propose a robust, i.e. well-posed, integrator for the conservative dynamical low-rank approach that conserves mass and momentum (up to machine precision) and significantly improves energy conservation. We also report improved qualitative results for some problems and show how the approach can be combined with a rank adaptive scheme.

翻译：动态低秩近似方法近年来已被证明在求解动力学方程时极为高效。然而，其主要缺陷在于无法保持底层物理问题的结构特性。例如，经典动态低秩方法会违反质量、动量及能量守恒定律。在文献[L. Einkemmer, I. Joseph, J. Comput. Phys. 443:110495, 2021]中，作者提出了一种守恒型动态低秩方法。但直接对由此导出的运动方程进行积分（与经典动态低秩方法类似）会导致方案不适定性。本文针对该守恒型动态低秩方法提出了一种鲁棒（即适定）的积分器，该方法可在机器精度范围内保持质量与动量守恒，并显著改善能量守恒性能。我们同时报告了部分问题上改进的定性结果，并展示了该方法与秩自适应方案的结合方式。