Convergence to equilibrium for discretizations of gradient-like flows on Riemannian manifolds.(English)Zbl 1299.65162

Summary: 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 by B. Merlet and M. Pierre [Commun. Pure Appl. Anal. 9, No. 3, 685–702 (2010; Zbl 1215.65132)] by showing that discrete solutions have the same behavior under mild hypotheses. In particular, we consider the $$\theta$$-scheme for systems with solutions in $$\mathbb {R}^d$$ 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.

MSC:

 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 58E20 Harmonic maps, etc. 65L12 Finite difference and finite volume methods for ordinary differential equations 35Q60 PDEs in connection with optics and electromagnetic theory

Zbl 1215.65132