The numerical integration of stiff equations is a challenging problem that needs to be approached by specialized numerical methods. Exponential integrators form a popular class of such methods since they are provably robust to stiffness and have been successfully applied to a variety of problems. The dynamical low- \rank approximation is a recent technique for solving high-dimensional differential equations by means of low-rank approximations. However, the domain is lacking numerical methods for stiff equations since existing methods are either not robust-to-stiffness or have unreasonably large hidden constants. In this paper, we focus on solving large-scale stiff matrix differential equations with a Sylvester-like structure, that admit good low-rank approximations. We propose two new methods that have good convergence properties, small memory footprint and that are fast to compute. The theoretical analysis shows that the new methods have order one and two, respectively. We also propose a practical implementation based on Krylov techniques. The approximation error is analyzed, leading to a priori error bounds and, therefore, a mean for choosing the size of the Krylov space. Numerical experiments are performed on several examples, confirming the theory and showing good speedup in comparison to existing techniques.

翻译：刚性方程的数值积分是一个具有挑战性的问题，需要采用专门的数值方法。指数积分器是一类广受欢迎的方法，因其被证明对刚性具有鲁棒性，并已成功应用于多种问题。动态低秩逼近是一种通过低秩近似求解高维微分方程的新技术。然而，该领域缺乏适用于刚性方程的数值方法，因为现有方法要么对刚性缺乏鲁棒性，要么隐含有不合理的大常数。本文聚焦于求解具有Sylvester-like结构的大规模刚性矩阵微分方程，这类方程具有良好的低秩近似性质。我们提出了两种新方法，具有优良的收敛特性、较小的内存占用和快速的计算性能。理论分析表明，新方法分别具有一阶和二阶精度。我们还提出了一种基于Krylov技术的实用实现方案。分析了近似误差，由此得出先验误差界，从而为Krylov空间维度的选择提供了依据。在多个算例上进行了数值实验，实验结果验证了理论分析，并显示出与现有技术相比良好的加速效果。