Gradient stability of high-order BDF methods and some applications. Issue 1 (2nd January 2020)

• Record Type: Journal Article
• Authors: Bouchriti, Anass; Pierre, Morgan; Alaa, Nour Eddine
• Is Part Of: Journal of difference equations and applications. Volume 26:Issue 1(2020)
• Page Start: 74
• Page End: 103
• Publication Date: 2020-01-02
• Subjects: BDF methods -- gradient flow -- Allen–Cahn equation -- Lojasiewicz–Simon inequality; 65Q10 -- 65P40 -- 37B25; Difference equations -- Periodicals; 515.625
• DOI: 10.1080/10236198.2019.1709062
• Languages: English
• ISSNs: 1023-6198

Abstract

Abstract : 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.