This paper presents a rank-adaptive implicit integrator for the tensor solution of three-dimensional diffusion and advection-diffusion equations. In particular, the recently developed Reduced Augmentation Implicit Low-rank (RAIL) integrator is extended from two-dimensional matrix solutions to three-dimensional tensor solutions stored in a Tucker tensor decomposition. Spectral methods are considered for spatial discretizations, and diagonally implicit Runge-Kutta (RK) and implicit-explicit (IMEX) RK methods are used for time discretization. The RAIL integrator first discretizes the partial differential equation fully in space and time. Then at each RK stage, the bases computed at the previous stages are augmented and reduced to predict the current (future) basis and construct projection subspaces. After updating the bases in a dimension-by-dimension manner, a Galerkin projection is performed by projecting onto the span of the previous bases and the newly updated bases. A truncation procedure according to a specified tolerance follows. Numerical experiments demonstrate the accuracy of the integrator using implicit and implicit-explicit time discretizations, as well as how well the integrator captures the rank of the solutions.

翻译：本文提出一种适用于三维扩散与对流-扩散方程张量解的自适应秩隐式积分器。具体而言，我们将最近发展的降维增强隐式低秩积分器从二维矩阵解推广至采用Tucker张量分解存储的三维张量解。空间离散采用谱方法，时间离散采用对角隐式龙格-库塔方法与隐式-显式龙格-库塔方法。该积分器首先对偏微分方程进行完整的时空离散，随后在每个RK阶段中，通过增广并约简先前阶段计算的基来预测当前（未来）基并构建投影子空间。在以逐维方式更新基后，通过投影至先前基与新更新基张成的空间进行伽辽金投影，随后根据指定容差执行截断操作。数值实验验证了该积分器在隐式及隐式-显式时间离散格式下的精度，并展示了其对解秩的捕捉能力。