  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 $L^2$-Wasserstein
metric.

Analysis of PDEs;Numerical Analysis