The use of variable grid BDF methods for parabolic equations leads to structures that are called variable (coefficient) Toeplitz. Here, we consider a more general class of matrix-sequences and we prove that they belong to the maximal $*$-algebra of generalized locally Toeplitz (GLT) matrix-sequences. Then, we identify the associated GLT symbols in the general setting and in the specific case, by providing in both cases a spectral and singular value analysis. More specifically, we use the GLT tools in order to study the asymptotic behaviour of the eigenvalues and singular values of the considered BDF matrix-sequences, in connection with the given non-uniform grids. Numerical examples, visualizations, and open problems end the present work.

翻译：变网格BDF方法在抛物型方程中的应用导致了一类称为变系数Toeplitz的结构。本文研究一类更一般的矩阵序列，并证明其属于广义局部Toeplitz矩阵序列的最大$*$-代数。随后，我们在一般情形及具体案例中识别了对应的GLT符号，并在两种情形下均给出了谱分析与奇异值分析。具体而言，我们运用GLT工具研究给定非均匀网格下所考虑BDF矩阵序列的特征值与奇异值的渐近行为。数值算例、可视化结果及开放性问题构成了本文的结尾。