Gradient stability of high-order BDF methods and some applications

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 th...

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Bibliographic Details
Published inJournal of difference equations and applications Vol. 26; no. 1; pp. 74 - 103
Main Authors Bouchriti, Anass, Pierre, Morgan, Alaa, Nour Eddine
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 02.01.2020
Taylor & Francis Ltd
Subjects
Allen-Cahn equation
BDF methods
Computer simulation
Gradient flow
Lojasiewicz-Simon inequality
Stability analysis
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Summary: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 . Some numerical simulations illustrate the theoretical results.
ISSN:1023-6198
1563-5120
DOI:10.1080/10236198.2019.1709062