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**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 |