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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Published in | Journal of difference equations and applications Vol. 26; no. 1; pp. 74 - 103 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Abingdon
Taylor & Francis
02.01.2020
Taylor & Francis Ltd |
Subjects | |
Online Access | Get full text |
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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. |
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ISSN: | 1023-6198 1563-5120 |
DOI: | 10.1080/10236198.2019.1709062 |