Stability of variable-step BDF2 and BDF3 methods

Abstract

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.

20 pages, 2 figures, 3 tables

References in corpus (1)

• Mesh-robustness of an energy stable BDF2 scheme with variable steps for the Cahn-Hilliard model

Cited by in corpus (4)

• $L^2$ norm error estimates of BDF methods up to fifth-order for the phase field crystal model
• Average energy dissipation rates of explicit exponential Runge-Kutta methods for gradient flow problems
• Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model
• Error estimation for numerical approximations of ODEs via composition techniques. Part II: BDF methods