  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.
