no. 1
A variational formulation of the BDF2 method for metric gradient flows
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 1, pp. 145-172

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.

DOI : 10.1051/m2an/2018045
Classification : 34G25, 35A15, 35G25, 35K46, 65L06, 65J08
Keywords: Gradient flow, second order scheme, BDF2, multistep discretization, minimizing movements, parabolic equations, nonlinear diffusion equations

Matthes, Daniel 1 ; Plazotta, Simon 1

1 
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Matthes, Daniel; Plazotta, Simon. A variational formulation of the BDF2 method for metric gradient flows. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 1, pp. 145-172. doi: 10.1051/m2an/2018045

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