This paper introduces a novel computational approach termed the Reduced Augmentation Implicit Low-rank (RAIL) method by investigating two predominant research directions in low-rank solutions to time-dependent partial differential equations (PDEs): dynamical low-rank (DLR), and step and truncation (SAT) tensor methods. The RAIL method, along with the development of the SAT approach, is designed to enhance the efficiency of traditional full-rank implicit solvers from method-of-lines discretizations of time-dependent PDEs, while maintaining accuracy and stability. We consider spectral methods for spatial discretization, and diagonally implicit Runge-Kutta (DIRK) and implicit-explicit (IMEX) RK methods for time discretization. The efficiency gain is achieved by investigating low-rank structures within solutions at each RK stage using a singular value decomposition (SVD). In particular, we develop a reduced augmentation procedure to predict the basis functions to construct projection subspaces. This procedure balances algorithm accuracy and efficiency by incorporating as many bases as possible from previous RK stages and predictions, and by optimizing the basis representation through SVD truncation. As such, one can form implicit schemes for updating basis functions in a dimension-by-dimension manner, similar in spirit to the K-L step in the DLR framework. We also apply a globally mass conservative post-processing step at the end of each RK stage. We validate the RAIL method through numerical simulations of advection-diffusion problems and a Fokker-Planck model, showcasing its ability to efficiently handle time-dependent PDEs while maintaining global mass conservation. Our approach generalizes and bridges the DLR and SAT approaches, offering a comprehensive framework for efficiently and accurately solving time-dependent PDEs with implicit treatment.

翻译：本文通过研究时间相关偏微分方程（PDE）低秩求解中的两个主要研究方向：动态低秩（DLR）方法与逐步截断（SAT）张量方法，提出了一种名为约化增广隐式低秩（RAIL）的新型计算方法。RAIL方法与SAT方法共同发展，旨在提升基于时间相关PDE的直线法离散化所得传统满秩隐式求解器的效率，同时保持精度与稳定性。我们采用谱方法进行空间离散化，并应用对角隐式龙格-库塔（DIRK）与隐式-显式（IMEX）RK方法进行时间离散化。通过利用奇异值分解（SVD）分析每个RK阶段解中的低秩结构，实现了效率提升。具体而言，我们开发了一种约化增广过程来预测基函数，从而构建投影子空间。该过程通过尽可能多地纳入先前RK阶段与预测产生的基函数，并通过SVD截断优化基表示，在算法精度与效率之间取得平衡。由此，可构建逐维度更新基函数的隐式格式，其思路与DLR框架中的K-L步骤类似。我们还在每个RK阶段结束时应用了全局质量守恒后处理步骤。通过对流-扩散问题与福克-普朗克模型的数值模拟，验证了RAIL方法在保持全局质量守恒的同时高效处理时间相关PDE的能力。我们的方法推广并桥接了DLR与SAT方法，为通过隐式处理高效且精确地求解时间相关PDE提供了一个综合框架。