Volume 53, Number 1, January–February 2019
|Page(s)||145 - 172|
|Published online||04 April 2019|
A variational formulation of the BDF2 method for metric gradient flows
Zentrum für Mathematik, Technische Universität München, 85747 Garching, Germany
* Corresponding author: firstname.lastname@example.org
Accepted: 17 July 2018
We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity – but no smoothness – of the augmented energy functional, we prove well-posedness of the method and convergence of the discrete approximations to a curve of steepest descent. In a smooth Hilbertian setting, classical theory would predict a convergence order of two in time, we prove convergence order of one-half in the general metric setting and under our weak hypotheses. Further, we illustrate these results with numerical experiments for gradient flows on a compact Riemannian manifold, in a Hilbert space, and in the L2-Wasserstein metric.
Mathematics Subject Classification: 34G25 / 35A15 / 35G25 / 35K46 / 65L06 / 65J08
Key words: Gradient flow / second order scheme / BDF2 / multistep discretization / minimizing movements / parabolic equations / nonlinear diffusion equations
© The authors. Published by EDP Sciences, SMAI 2019
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