  We prove that the two-step backward differentiation formula (BDF2) method is
stable on arbitrary time grids; while the variable-step BDF3 scheme is stable
if almost all adjacent step ratios are less than 2.553. These results relax the
severe mesh restrictions in the literature and provide a new understanding of
variable-step BDF methods. Our main tools include the discrete orthogonal
convolution kernels and an elliptic-type matrix norm.
