Sylvester-Preconditioned Adaptive-Rank Implicit Time Integrators for Advection-Diffusion Equations with Variable Coefficients

We consider the adaptive-rank integration of {2D and 3D} time-dependent advection-diffusion partial differential equations (PDEs) with variable coefficients. We employ a standard finite-difference method for spatial discretization coupled with diagonally implicit Runge-Kutta temporal schemes. The discrete equation is a generalized Sylvester equation (GSE), which we solve with an adaptive-rank algorithm structured around three key strategies: {(i) constructing dimension-wise subspaces based on an extended Krylov strategy, (ii) developing an effective preconditioner for the reduced coefficient matrix, and (iii) efficiently computing the residual of the equation without explicitly reverting to the full-rank form. {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.} The computational complexity of the proposed approach {is demonstrated numerically to} be comparable to the constant-coefficient case [El Kahza et al, J. Comput. Phys., 518 (2024)], scaling as $\mathcal{O}(N {r^2} + {r^{d+1}})$ for $d$-dimensional problems (here, $d = 2$ or $3$), with $N$ the resolution in one dimension and $r$ the maximal rank during the Krylov iteration (which we find to be largely independent of $N$). We present numerical examples that illustrate the computational efficacy and complexity of our algorithm.}