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MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
25/2021

Numerical attractors for rough differential equations

Hoang Duc Luu and Peter Kloeden

Abstract

We study the explicit Euler scheme to approximate the solutions of rough differential equations under a bounded or linear diffusion term, where the drift term satisfies a local Lipschitz continuity and a bounded linear growth condition. The Euler scheme is then proved to converge for a given solution, although the approximation of the error depends on the initial condition. For a dissipative drift term with linear growth condition and a bounded diffusion term, the numerical solution under a regular grid generates a random dynamical system which admits a random pullback attractor. We also prove that for bounded drift and diffusion terms, the numerical pullback attractor then converges upper semi-continuously to the continuous-time pullback attractor as the time step goes to zero.

Received:
25.10.21
Published:
25.10.21
Keywords:
rough differential equations, Euler scheme, random dynamical systems, random attractors

Related publications

inJournal
2023 Repository Open Access
Hoang Duc Luu and Peter E. Kloeden

Numerical attractors for rough differential equations

In: SIAM journal on numerical analysis, 61 (2023) 5, pp. 2381-2407