Abstract
The existence of numerical attractors for lattice dynamical systems is established, where the implicit Euler scheme is used for time discretisation. Infinite dimensional discrete lattice systems as well as their finite dimensional truncations are considered. It is shown that the finite dimensional numerical attractors converge upper semicontinuously to the global attractor of the original lattice model as the discretisation step size tends to zero.
Highlights
1 Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA 2 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 3 IMAPP - Mathematics, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands Journal of Dynamics and Differential Equations (2020) 32:1457–1474 leads to a lattice dynamical system (LDS)
The goal of this work is to investigate the existence of numerical attractors for the LDS (1), where the implicit Euler scheme (IES) is used for time-discretisation
We show that the numerical attractors converge upper semicontinuously to the global attractor for the continuous time lattice model as the discretisation step size tends to zero
Summary
Using a finite difference quotient to discretise the Laplace operator in a reaction–diffusion equation defined on the real line R,. The existence and upper semicontinuity of global attractors for the LDS (1) and its finite dimensional truncations were studied by Bates et al [2] in the Hilbert space 2 of square summable bi-infinite real-valued sequences. The goal of this work is to investigate the existence of numerical attractors for the LDS (1), where the implicit Euler scheme (IES) is used for time-discretisation. We show that the numerical attractors converge upper semicontinuously to the global attractor for the continuous time lattice model as the discretisation step size tends to zero. The upper semicontinuous convergence of the finite dimensional numerical attractors to the global attractor of the original lattice system (1) follows. We consider finite dimensional truncations of the discretised lattice system and show that their numerical attractors converge upper semicontinuously to the global attractor of the original lattice model (1).
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