Gradient stability of high-order BDF methods and some applications
Anass Bouchriti, Morgan Pierre, Nour Eddine Alaa · The Journal of Difference Equations and Applications · 2020
It is well-known that the backward differentiation formulae (BDF) of order 1, 2 and 3 are gradient stable. This means that when such a method is used for the time discretization of a gradient flow, the associated discrete dynamical system exhibit properties similar to the continuous case, such as the existence of a Lyapunov functional. By means of a Lojasiewicz–Simon inequality, we prove convergence to equilibrium for the 3-step BDF scheme applied to the Allen–Cahn equation with an analytic nonlinearity. By introducing a notion of quadratic-stability, we also show that the BDF methods of order 4 and 5 are gradient stable, and that the k-step BDF schemes are not gradient stable for k≥7. Some numerical simulations illustrate the theoretical results.