We consider the adaptive-rank integration of general time-dependent advection-diffusion partial differential equations (PDEs) with spatially variable coefficients. We employ a standard finite-difference method for spatial discretization coupled with diagonally implicit Runge-Kutta schemes for temporal discretization. The fully discretized scheme can be written as a generalized Sylvester equation, 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 Averaged-Coefficient Sylvester (ACS) preconditioner to invert the reduced system for the coefficient matrix efficiently, and (iii) efficiently computing the residual of the equation without explicitly reverting to the full-rank form. The proposed approach features a computational complexity of O(Nr2 + r3), where r represents the rank during the Krylov iteration and N is the resolution in one dimension, commensurate with the constant-coefficient case [El Kahza et al, J. Comput. Phys., 518 (2024)]. We present numerical examples that illustrate the computational efficacy and complexity of our algorithm.

翻译：本文研究具有空间变系数的一般时间依赖对流扩散偏微分方程的自适应秩积分方法。我们采用标准有限差分法进行空间离散化，并结合对角隐式Runge-Kutta格式进行时间离散化。全离散格式可表述为广义Sylvester方程，我们通过围绕三个核心策略构建的自适应秩算法进行求解：(i) 基于扩展Krylov策略构建维度方向子空间，(ii) 开发高效的均值系数Sylvester预条件子以快速求取约简系统的系数矩阵逆，(iii) 无需显式恢复至满秩形式即可高效计算方程残量。所提方法具有O(Nr² + r³)的计算复杂度，其中r表示Krylov迭代过程中的秩，N为一维分辨率，该复杂度与常系数情形相当[El Kahza et al, J. Comput. Phys., 518 (2024)]。我们通过数值算例展示了算法的计算效能与复杂度特性。