A Reduced Augmentation Implicit Low-rank (RAIL) integrator for solving three-dimensional diffusion and advection-diffusion equations in the Tucker tensor format

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.