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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 ℝ¹. Firstly, we prove that this semi-discrete equation provides a discrete infinite-dimensional dynamical system in H¹(ℝ¹). Then we prove that this system possesses a global attractor ${}_{\tau }$ in H¹(ℝ¹). In addition, we show that the global attractor ${}_{\tau }$ is regular, i.e., ${}_{\tau }$ is actually...