In this paper, we consider discretizations of systems of disfferential 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 θ-scheme for systems with solutions in Rd and a projected θ-scheme for systems de ned 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.
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