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Abstract

This paper is concerned with the study of the large time behavior and especially the regularity of the global attractor for the semi-discrete in time Crank–Nicolson scheme to discretize the Benjamin–Bona–Mahony equation on $\mathbb {R}^1$. Firstly, we prove that this semi-discrete equation provides a discrete infinite-dimensional dynamical system in $H^1(\mathbb {R}^1)$. Then we prove that this system possesses a global attractor $\mathcal {A}_\tau $ in $H^1(\mathbb {R}^1)$. In addition, we show that the global attractor $\mathcal {A}_\tau $ is regular, i.e., $\mathcal {A}_\tau $ is actually included, bounded and compact in $H^2(\mathbb {R}^1)$. Finally, we estimate the finite fractal dimensions of $\mathcal {A}_\tau $.