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Maximum time step for high order BDF methods applied to gradient flows

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Abstract

For a backward differentiation formula (BDF) applied to the gradient flow of a semiconvex function, quadratic stability implies gradient stability. Namely, it is possible to build a Lyapunov functional for the discrete-in-time dynamical system, with a restriction on the time step. The maximum time step which can be derived from quadratic stability has previously been obtained for the BDF1, BDF2 and BDF3 schemes. Here, we compute it for the BDF4 and BDF5 methods. We also prove that the BDF6 scheme is not quadratically stable. Our results are based on the tools developed by Dahlquist and other authors to show the equivalence of A-stability and G-stability. We give several applications of gradient stability to the asymptotic behaviour of sequences generated by BDF schemes.

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Acknowledgements

The author is thankful to Michel Pierre for suggesting him to investigate in detail the quadratic stability of the BDF6 scheme.

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  1. Laboratoire de Mathématiques et Applications, Université de Poitiers, CNRS, 86073, Poitiers, France

    Morgan Pierre

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Pierre, M. Maximum time step for high order BDF methods applied to gradient flows. Calcolo 59, 36 (2022). https://doi.org/10.1007/s10092-022-00479-0

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