The two-step backward difference formula (BDF) method on variable grids for parabolic equations with self-adjoint elliptic part is considered. Standard stability estimates for adjacent time-step ratios \(r_j:=k_j/k_{j-1}\leqslant 1.8685\) and 1.9104, respectively, have been proved by Becker (BIT 38:644–662, 1998) and Emmrich (J Appl Math Comput 19:33–55, 2005) by the energy technique with a single multiplier. Even slightly improving the ratio is cumbersome. In this paper, we present a novel technique to examine the positive definiteness of banded matrices that are neither Toeplitz nor weakly diagonally dominant; this result can be viewed as a variant of the Grenander–Szegő theorem. Then, utilizing the energy technique with two multipliers, we establish stability for adjacent time-step ratios up to 1.9398.
