A low-rank, high-order implicit-explicit integrator for three-dimensional convection-diffusion equations

Abstract

This paper presents a rank-adaptive implicit-explicit integrator for the tensor approximation of three-dimensional convection-diffusion equations. In particular, the recently developed Reduced Augmentation Implicit Low-rank (RAIL) integrator is extended from the two-dimensional matrix case to the three-dimensional tensor case. The solutions are approximated using a Tucker tensor decomposition. The RAIL integrator first discretizes the partial differential equation fully in space and time using traditional methods. Here, spectral methods are considered for spatial discretizations, and implicit-explicit Runge-Kutta (IMEX RK) methods are used for time discretization. At each RK stage: the bases computed at the previous stages are augmented and reduced to construct projection subspaces. After updating the bases in a dimension-by-dimension manner, a Galerkin projection is performed to update the coefficients stored in the core tensor. As such, the algorithm balances high-order accuracy from spanning as many bases as possible from previous stages, with efficiency from leveraging low-rank structures in the solution. A post-processing step follows to maintain a low-rank solution while conserving mass, momentum, and energy. We validate the proposed method on a number of convection-diffusion problems, including a Fokker-Planck model, and a 3d viscous Burgers' equation.