Differential and Integral Equations

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

Benoît Merlet and Thanh Nhan Nguyen

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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.

Article information

Source
Differential Integral Equations, Volume 26, Number 5/6 (2013), 571-602.

Dates
First available in Project Euclid: 14 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1363266079

Mathematical Reviews number (MathSciNet)
MR3086400

Zentralblatt MATH identifier
1299.65162

Subjects
Primary: 65L06: Multistep, Runge-Kutta and extrapolation methods 65L20: Stability and convergence of numerical methods 65P05

Citation

Merlet, Benoît; Nguyen, Thanh Nhan. Convergence to equilibrium for discretizations of gradient-like flows on Riemannian manifolds. Differential Integral Equations 26 (2013), no. 5/6, 571--602. https://projecteuclid.org/euclid.die/1363266079


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