Convergence to equilibrium for discretizations of gradient-like flows on Riemannian manifolds

Abstract

In this paper, we consider discretizations of systems of differential equations on manifolds that admit a strict Lyapunov function. We study the long-time behavior of the discrete solutions. In the continuous case, if a solution admits an accumulation point for which a Lojasiewicz inequality holds then its trajectory converges. Here we continue the work started in [18] by showing that discrete solutions have the same behavior under mild hypotheses. In particular, we consider the $\theta$-scheme for systems with solutions in $\mathbf{R}^d$ and a projected $\theta$-scheme for systems defined on an embedded manifold. As illustrations, we show that our results apply to existing algorithms: 1) Alouges' algorithm for computing minimizing discrete harmonic maps with values in the sphere, and 2) a discretization of the Landau--Lifshitz equations of micromagnetism.