Sylvester-preconditioned adaptive-rank implicit time integrators for advection-diffusion equations with variable coefficients

Abstract

We consider the adaptive-rank integration of multi-dimensional time-dependent advection-diffusion partial differential equations (PDEs) with variable coefficients. We employ a standard finite-difference method for spatial discretization coupled with high-order diagonally implicit Runge-Kutta temporal schemes. The discrete equation is a generalized Sylvester equation (GSE), which we solve with a projection-based adaptive-rank algorithm structured around two key strategies: (i) constructing dimension-wise subspaces using a novel atypical extended Krylov strategy, and (ii) efficiently solving the basis coefficient matrix with a preconditioned GMRES solver. The low-rank decomposition is performed in 2D using SVD and with high-order SVD (HOSVD) in 3D to represent the tensor in a compressed Tucker format. For $d$-dimensional problems (here, $d=2$ or 3), the computational complexity and memory storage of the approach are found numerically to scale as $O\left(N{r}^2\right)+O\left(r^{d+1}\right)$ and $O\left(Nr\right)+O\left(r^d\right)$, respectively, with $N$ the one-dimensional resolution and $r$ the maximal rank during the Krylov iteration (which we find to be largely independent of $N$ on our numerical examples). We present numerical examples that illustrate the advertised properties of the algorithm.